You will be provided with a reference and some statements. Please determine whether each statement is 'supported', 'unsupported', or 'unknown' with respect to the reference. Please note:
First, assess whether the reference contains any valid content. If the reference contains no valid information, such as a 'page not found' message, then all statements should be considered 'unknown'.
If the reference is valid, for a given statement: if the facts or data it contains can be found entirely or partially within the reference, it is considered 'supported' (data accepts rounding); if all facts and data in the statement cannot be found in the reference, it is considered 'unsupported'.

You should return the result in a JSON list format, where each item in the list contains the statement's index and the judgment result, for example:
[
    {
        "idx": 1,
        "result": "supported"
    },
    {
        "idx": 2,
        "result": "unsupported"
    }
]

Below are the reference and statements:
<reference>
arXiv:1902.07449v1 [q-fin.PM] 20 Feb 2019

Robust Asset Allocation for Robo-Advisors∗
Thibault Bourgeron
Quantitative Research
Amundi Asset Management, Paris
thibault.bourgeron@amundi.com

Edmond Lezmi
Quantitative Research
Amundi Asset Management, Paris
edmond.lezmi@amundi.com

Thierry Roncalli
Quantitative Research
Amundi Asset Management, Paris
thierry.roncalli@amundi.com
September 2018
Abstract
In the last few years, the financial advisory industry has been impacted by the
emergence of digitalization and robo-advisors. This phenomenon affects major financial
services, including wealth management, employee savings plans, asset managers, private
banks, pension funds, banking services, etc. Since the robo-advisory model is in its early
stages, we estimate that robo-advisors will help to manage around $1 trillion of assets
in 2020 (OECD, 2017). And this trend is not going to stop with future generations,
who will live in a technology-driven and social media-based world.
In the investment industry, robo-advisors face different challenges: client profiling,
customization, asset pooling, liability constraints, etc. In its primary sense, roboadvisory is a term for defining automated portfolio management. This includes automated trading and rebalancing, but also automated portfolio allocation. And this
last issue is certainly the most important challenge for robo-advisory over the next five
years. Today, in many robo-advisors, asset allocation is rather human-based and very
far from being computer-based. The reason is that portfolio optimization is a very
difficult task, and can lead to optimized mathematical solutions that are not optimal
from a financial point of view (Michaud, 1989). The big challenge for robo-advisors is
therefore to be able to optimize and rebalance hundreds of optimal portfolios without
human intervention.
In this paper, we show that the mean-variance optimization approach is mainly
driven by arbitrage factors that are related to the concept of hedging portfolios. This
is why regularization and sparsity are necessary to define robust asset allocation. However, this mathematical framework is more complex and requires understanding how
norm penalties impacts portfolio optimization. From a numerical point of view, it also
requires the implementation of non-traditional algorithms based on ADMM methods
and proximal operators.

Keywords: Robo-advisor, asset allocation, active management, portfolio optimization,
Black-Litterman model, spectral filtering, machine learning, Tikhonov regularization, mixed
penalty, ridge regression, lasso method, sparsity, ADMM algorithm, proximal operator.
JEL classification: C61, C63, G11.
∗ The authors are very grateful to Silvia Bocchiotti, Arnaud Gamain, Patrick Herfroy, Matthieu Keip,
Didier Maillard, Hassan Malongo, Binh Phung-Que, Christophe Romero and Takaya Sekine for their helpful
comments.

1

Robust Asset Allocation for Robo-Advisors

1

Introduction

The concept of portfolio optimization has a long history and dates back to the seminal work
of Markowitz (1952). In this paper, Markowitz defined precisely what portfolio selection
means: “the investor does (or should) consider expected return a desirable thing and variance
of return an undesirable thing”. This was the starting point of mean-variance optimization
and portfolio allocation based on quantitative models. In particular, the Markowitz approach
became the standard model for strategic asset allocation until the end of the 2000s.
Since the financial crisis of 2008, another model has emerged and is now a very serious
contender for asset allocation (Roncalli, 2013). The risk budgeting approach is successfully
used for managing multi-asset portfolios, equity risk factors or alternative risk premia. The
main difference with mean-variance optimization is the objective function. The Markowitz
approach mainly focuses on expected returns and exploits the trade-off between performance
and volatility. The risk budgeting approach is based on the risk allocation of the portfolio,
and does not take into account expected returns of assets.
The advantage of the risk budgeting approach is that it produces stable and robust portfolios. On the contrary, mean-variance optimization is very sensitive to input parameters.
These stability issues make the practice of portfolio optimization less attractive than the
theory (Michaud, 1989). Even for strategic asset allocation, many weight constraints need
to be introduced in order to regularize the mathematical solution and obtain an acceptable
financial solution. In the case of tactical asset allocation, professionals generally prefer to
implement the model of Black and Litterman (1991, 1992), because the optimized portfolio
depends on the current allocation. Therefore, the Black-Litterman model appears to be
slightly more robust than the Markowitz model because having a benchmark or introducing
a tracking error constraint is already a form of portfolio regularization. However, since the
Black-Litterman model is a slight modification of the Markowitz model, it suffers from the
same drawbacks.
Since the 1990s, academics have explored how to robustify portfolio optimization in
two different directions. The first one deals with the estimation of the input parameters.
For instance, we can use de-noising methods (Laloux et al., 1999) or shrinkage approaches
(Ledoit and Wolf, 2004) to reduce estimation errors of the covariance matrix. The second
one deals with the objective function. As explained by Roncalli (2013), the Markowitz model
is an aggressive model of active management due to the mean-variance objective function.
Academics have suggested regularizing the optimization problem by adding penalization
functions. For instance, it is common to include a L1 or L2 norm loss function. The
advantage of this is to obtain a “sparser” or “smoother” solution.
The success of risk parity, equal risk contribution (ERC) and risk budgeting portfolios has
put these new developments in second place. However, the rise of robo-advisors is changing
the current trend and highlights the need for active allocation models that are focused on
expected returns. Indeed, the challenge of robo-advice concerns tactical asset allocation and
not the portfolio construction of strategic asset allocation. Building a defensive, balance or
dynamic portfolio profile is not an issue, because they are defined from an ex-ante point
of view. Quantitative models can be used to define this step, but they are not necessarily
required. For example, this step can also be done using a discretionary approach, since
portfolio profiles are revised once and for all. The difficulty lies with the life of the invested
portfolio and the dynamic allocation. A robo-advisor that would consist in rebalancing a
constant-mix allocation is not a true robo-advisor, since it is reduced to the profiling of
clients. The main advantage of robo-advisors is to perform dynamic allocation by including
investment views, side assets or the client’s dynamic constraints, or some alpha engines
provided by the robo-advisor’s manager or distributor.
2

Robust Asset Allocation for Robo-Advisors

The challenge for a robo-advisor is therefore to perform dynamic allocation or tactical
asset allocation in a systematic way without human interventions. In this case, expected
returns or trading signals must be taken into account. One idea is to consider an extension
of the ERC portfolio by using a risk measure that depends on expected returns (Roncalli,
2015). However, this approach is not always suitable when we target a high tracking error.
Otherwise, it makes a lot of sense for the mean-variance optimization to be the allocation
engine of robo-advisors. As said previously, the challenge is to develop a robust asset allocation model. The purpose of this research is to provide a practical solution that does not
require human interventions.
This paper is organized as follows. Section Two illustrates the practice of mean-variance
optimization and highlights the limits of such models. In Section Three, we apply the theory
of regularization to asset allocation. In particular, we point out the calibration procedure
of the Lagrange coefficients of norm functions. In Section Four, we consider application to
robo-advisory. Finally, Section Five offers some concluding remarks.

2

Practice and limits of mean-variance optimization

2.1

The mean-variance optimization framework

We follow the presentation of Roncalli (2013). We consider a universe of n assets. Let
x = (x1 , . . . , xn ) be the vector of weights in the portfolio. We denote by µ and Σ the vector
of expected returns and the covariance matrix of asset returns. It follows that the expected
√
return and the volatility of the portfolio are equal to µ (x) = x> µ and σ (x) = x> Σx.
The Markowitz approach consists in maximizing the expected return of the portfolio under
a volatility constraint (σ-problem):
x? = arg max µ (x) s.t. σ (x) ⩽ σ ?

(1)

or minimizing the volatility of the portfolio under a return constraint (µ-problem):
x? = arg min σ (x) s.t. µ (x) ⩾ µ?

(2)

Replacing the volatility by the variance scaled with the factor 1/2 does not change the
solution. Therefore, we deduce that the Lagrange functions associated with Problems (1)
and (2) are:


1 2
1 2
L1 (x, λ1 , σ ? ) = x> µ − λ1
σ (x) − σ ?
2
2
and:

1 2
σ (x) − λ2 (µ (x) − µ? )
2
They satisfy L2 (x, λ2 , 0) = −λ2 L1 (x, θ, 0) where θ = λ−1
is the risk aversion of the
2
quadratic utility function. As strong duality holds, these two problems are equivalent.
Moreover, we can show that they can be written as a standard quadratic programming
problem (Markowitz, 1956):
L2 (x, λ2 , µ? ) =

1
x? (γ) = arg min x> Σx − γx> µ
2

(3)

where γ is the risk/return trade-off parameter. Since the problem is strongly convex and
the solution is x? (γ) = γΣ−1 µ, we deduce that the solution of the µ-problem is given by:
γ=

µ?
µ> Σ−1 µ
3

Robust Asset Allocation for Robo-Advisors

whereas the solution of the σ-problem is obtained for the following value of γ:
σ?
γ=p
µ> Σ−1 µ
The previous framework can be extended by considering a risk-free asset and portfolio
constraints:
x? (γ)

1
arg min x> Σx − γx> (µ − r1)
2
s.t. x ∈ Ω
=

(4)

where r is the risk-free rate and Ω is the set of restrictions. Let µ− ⩽ µ (x) ⩽ µ+ and
σ − ⩽ σ (x) ⩽ σ + be the bounds of the expected return and the volatility such that x ∈ Ω.
It follows that there is a solution to the σ-problem and the µ-problem if σ ? ⩾ σ − and
µ? ⩽ µ+ .
Remark 1 The Sharpe ratio is the standard risk/return measure used in finance, and corresponds to the zero-homogeneous quantity:
SR (x | r)

=
=

µ (x) − r
σ (x)
x> µ − r
√
x> Σx

The capital asset pricing model (CAPM) defines the tangency portfolio as the optimized
portfolio
Pn that has the maximum Sharpe ratio. When the capital budget is reached (meaning
that i=1 xi = 1), the solution of Problem (4) is equal to x? = γΣ−1 (µ − r1) where γ ? =
−1
1> Σ−1 (µ − r1)
. Since the matrix Σ has a unique symmetric positive definite square
1/2
root denoted by Σ , the Cauchy-Schwarz inequality yields:
2

2 

>
x> (µ − r1) = x> Σ1/2 Σ−1/2 (µ − r1) ⩽ x> Σx (µ − r1) Σ−1 (µ − r1)
The equality holds if and only if there exists a scalar γ ∈ R such that Σ1/2 x = γΣ−1/2 (µ − r1).
It follows that:
q
∀ x ∈ Rn

>

(µ − r1) Σ−1 (µ − r1)

SR (x | r) ⩽

(5)

We deduce that the set of portfolios maximizing the Sharpe ratio is the one-dimensional
vector space defined by x ∈ Σ−1 (µ − r1). This means that unconstrained and constrained
portfolio optimizations are related when we impose only one simple constraint like the capital
budget restriction. In more complex cases, the constrained solution is not necessarily related
to the unconstrained solution. However, the bound remains valid, because it only depends on
the Cauchy-Schwarz inequality.
The previous result highlights the importance of constraints in portfolio optimization.
A portfolio is long-only if ∀ i ∈ {1, . . . , n} xi ⩾ 0 whereas it is long-short if ∃ (i, j) ∈
{1, . . . , n} such that xi > 0 and xj < 0. For long-only portfolios,
Pn a capital budget is
usually assumed, meaning that the portfolio is fully invested ( i=1 xi = 1). For longshort portfolios, professionals sometimes impose a neutral orP
zero-capital budget, implying
n
that the long exposure is financed
by
the
short
exposure
(
i=1 xi = 0). They can also
Pn
impose leverage constraints ( i=1
|x
|
⩽
c),
while
risk-budgeting
portfolios require adding
i
Pn
a logarithmic barrier constraint ( i=1 ωi ln xi ⩾ c).
4

Robust Asset Allocation for Robo-Advisors

In practice, the quantities µ and Σ are unknown and must be specified. We can assume
that they are estimated using an historical sample {R1 , . . . , RT } where Rt is the vector of
asset returns at time t. Let µ̂ and Σ̂ be the corresponding estimators. We have:
µ̂ =

T
X

wt Rt

t=1

and:
Σ̂ =

T
X

wt (Rt − µ̂) (Rt − µ̂)

>

t=1

PT
where wt is the weighting scheme such that t=1 wt = 1. In Appendix A.2 on page 46, we
show that Problem (4) can be written as follows1 :
x? (γ)

=

arg min


1
2
kRxkW − γx> R> w − r1
2

(6)

s.t. x ∈ Ω
where w = (w1 , . . . , wT ) ∈ RT , R = (R1 , . . . , RT ) ∈ RT ×n and W = diag (w) − ww> . In
this case, the Markowitz solution is the portfolio that maximizes the backtest for a given
volatility. When wt+1 ⩾ wt , we conclude that Problem (4) is a trend-following optimization
program, whose moving average is defined by the weighting scheme w. In order not to be
trend-following, we have to use a vector of expected returns µ that does not satisfy wt+1 ⩾ wt
or that does not depend on the sample of asset returns.

2.2

Stability issues

According to Hadamard (1902), a well-posed problem must satisfy three properties:
1. a solution exists;
2. the solution is unique;
3. the solution’s behavior changes continuously with the initial conditions.
We recall that the solution to Problem (3) is x? (γ) = γΣ−1 µ. If Σ has no zero eigenvalues,
it follows that the existence and uniqueness is ensured, but not necessarily the stability.
Indeed, this third property implies that Σ has no “small” eigenvalues. This problem is
extensively illustrated by Bruder et al. (2013) and Roncalli (2013). If we consider the
eigendecomposition Σ = V ΛV > , we have Σ−1 = V Λ−1 V > and x? (γ) = γV Λ−1 V > µ. It
follows that V > x? (γ) = γΛ−1 V > µ or:
x̃? ∝ Λ−1 µ̃

(7)

where x̃? = V > x? (γ) and µ̃ = V > µ. By applying the change of basis V −1 , we notice that
the Markowitz solution is proportional to the vector of return and inversely proportional to
the eigenvectors. We conclude that the mean-variance optimization problem mainly focuses
on the small eigenvalues. This is why the stability property is lacking in the original portfolio
optimization problem.
Let us consider an example to illustrate this problem. The investment universe is composed of 4 assets. The expected returns are equal to µ1 = 7%, µ2 = 8%, µ3 = 9% and
1 The norm kxk is equal to
A

1/2
x> Ax
. All the notations are defined in Appendix A.1 on page 46.

5

Robust Asset Allocation for Robo-Advisors

µ4 = 10% whereas the volatilities are equal to σ1 = 15%, σ2 = 18%, σ3 = 20% and
σ4 = 25%. The correlation matrix is the following:


1.00
 0.50 1.00


C=
 0.50 0.50 1.00

0.60 0.50 0.40 1.00
The portfolio manager’s objective is to maximize the expected return for a 15% volatility
target and a full investment2 . The optimal portfolio x? is (26.3%, 25.5%, 32.3%, 15.9%). In
Table 1, we indicate how this solution differs when we slightly change the value of input
parameters. For example, if the volatility of the third asset is equal to 19%, the weight
of the third asset becomes 39.1% instead of 32.3%. In real life, we know exactly the true
parameters. For instance, there is a low probability that the realized correlation matrix is
exactly the one specified above. If we consider a uniform correlation matrix of 70%, we
observe significant differences in terms of allocation.
Table 1: Sensitivity of the MVO portfolio to input parameters
σ3
C
µ2
x1
x2
x3
x4

19%

26.30
25.52
32.28
15.90

21.48
22.90
39.10
16.52

21%

30.20
27.79
26.48
15.53

C4 (30%)

C4 (70%)

7.03
24.23
37.53
31.21

54.59
26.81
22.38
−3.78

5%
54.72
−2.43
35.38
12.34

21%
C4 (70%)
7%
70.75
13.95
16.57
−1.27

We have seen that the lack of stability is due to the small eigenvalues of the covariance
matrix. More specifically, we notice that the important quantity in mean-variance optimization is not the covariance matrix itself, but the precision matrix, which is the inverse of the
covariance matrix. In Tables 2 and 3, we have reported the eigendecomposition of Σ and
I = Σ−1 . We verify that the eigenvectors of the precision matrix are the same as those
of the covariance matrix, but the eigenvalues of the precision matrix are the inverse of the
eigenvalues of the covariance matrix. This means that the risk factors are the same, but
Table 2: Principal component analysis of the covariance matrix Σ
Factor
1
2
Asset
3
4
Eigenvalue
% cumulated

1
36.16%
42.19%
44.74%
70.08%
0.10%
63.80%

2
2.44%
25.48%
73.10%
−63.26%
0.03%
18.72%

3
5.72%
−86.21%
46.52%
19.25%
0.02%
10.65%

4
−93.03%
11.76%
22.16%
26.76%
0.01%
6.83%

they are in reverse order. We see that the most important risk factor for portfolio optimization is a long/short portfolio, which is short on the first asset and long on the other assets.
The second most important risk factor is another long/short portfolio, which is short on
2 We only impose that the sum of the weights is equal to 100%.

6

Robust Asset Allocation for Robo-Advisors

Table 3: Eigendecomposition of the precision matrix I
Factor
1
2
Asset
3
4
Eigenvalue
% cumulated

1
−93.03%
11.76%
22.16%
26.76%
93.06%
47.33%

2
5.72%
−86.21%
46.52%
19.25%
59.65%
30.34%

3
2.44%
25.48%
73.10%
−63.26%
33.94%
17.26%

4
36.16%
42.19%
44.74%
70.08%
9.96%
5.06%

the second asset and long on the third asset3 . Any changes in the covariance matrix then
impacts the largest eigenvalues of I and the long/short risk factors.

2.3

Which risk factors are important?

The previous eigendecomposition analysis is the traditional way to illustrate the stability
issue (Roncalli, 2017). However, the corresponding arbitrage factors are difficult to interpret
and, moreover, they do not fully help understand the Markowitz machinery, in particular
how mean-variance portfolios are built. In this section, we use the method developed by
Stevens (1998) in order to better characterize the underlying mechanism.
We have seen that the solution is x? (γ) = γΣ−1 µ. If we assume that asset returns are
independent – C = In , we obtain the famous result:
µi
x?i (γ) = γ 2
σi
The optimal weights are proportional to expected returns and inversely proportional to
variances of asset returns. In the general case – C 6= In , Stevens (1998) shows that the
optimal portfolio x? is connected to the linear regression4 :
(−i)

Ri,t = αi + βi> Rt

+ εi,t

(8)

(−i)

where Rt
denotes the vector of asset returns excluding the ith asset. By noting R2i the
coefficient of determination and s2i the variance of εi,t , we have:
 −1 
Σ i,i =
and:
 −1 
Σ i,j = −

1
σi2 (1 − R2i )

βi,j
=− 2
2
σi (1 − R2i )
σj

βj,i

1 − R2j

We deduce that:
x?i (γ)



= γ Σ−1 µ i
= γ

µi − βi> µ(−i)
σi2 (1 − R2i )

3 On Page 63, we have reported the representation quality and the contribution of each variable for the
PCA factors of Σ. Since the second risk factor of I is the third risk factor of Σ, we deduce that the first and
fourth assets have a very small contribution (respectively 0.33% and 3.71%).
4 This means that:
X
Ri,t = αi +
βi,j Rt,j + εi,t
j6=i

7

Robust Asset Allocation for Robo-Advisors

where µ(−i) is the vector of expected returns excluding the ith asset. Since we have5 s2i =
σi2 1 − R2i and αi = µi − βi> µ(−i) , we obtain:
αi
x?i (γ) = γ 2
si
In the general case, the optimal weights are proportional to idiosyncratic returns αi and
inversely proportional to idiosyncratic variances s2i .
We notice that βi represents the best portfolio for replicating the returns of Asset i. This
is why it is called the hedging (or tracking) portfolio of Asset i. The idiosyncratic return αi
is the difference between the expected return µi of Asset i and the expected return βi> µ(−i)
of its hedging portfolio. The idiosyncratic volatility si is the standard deviation of residuals
εi,t . It is also equal to the volatility of the tracking errors ei,t = Ri,t − R̂i,t where R̂i,t is the
return of the hedging portfolio. The hedging portfolio concept is at the core of the Markowitz
optimization. Indeed, the Markowitz framework consists in estimating the hedging strategy
βi for each asset, and in forming two portfolios:
1. the first portfolio y ? is the optimal portfolio of assets assuming that assets are not
correlated:
µi
yi? = γ 2
σi
2. the second portfolio z ? is the optimal portfolio of the hedging strategies:6 :
zi? = γ

βi> µ(−i)
σi2 − s2i

We deduce that:
x?i (γ)


βi> µ(−i)
− γ 2
=
σi (1 − R2i )

 

1
γµi
σi2 − s2i
γβi> µ(−i)
=
·
−
·
(1 − R2i ) σi2
σi2 (1 − R2i ) σi2 − s2i




µi
β > µ(−i)
= (1 + ωi ) φ−1 2 − ωi φ−1 i2
σi
σi − s2i
= yi? + ωi (yi? − zi? )

γ

µi
2
σi (1 − R2i )

where:
ωi =





R2i
σi2 − s2i
=
1 − R2i
s2i

To take into account the correlation diversification, the optimal portfolio x? adds to the
portfolio y ? a long/short exposure between y ? and z ? with a leverage that depends on the
quality of the hedge.
Let us consider the previous example. In Table 4, we have reported the linear regressions
between the four assets, which are the hedging portfolios of each asset. We observe that the
coefficient of determination lies between 33.5% and 45.8%. R2i is the highest for the first
asset, because it exhibits the largest cross-correlations. Therefore, it is the lowest contributor
to the diversification whereas the third asset is the highest contributor to the diversification.
5 See Appendix A.3 on page 47.
6 Because the hedging strategies are independent and we have var

σi2 − s2i .

8



(−i)
βi> Rt
= var (Ri,t ) − var (εi,t ) =

Robust Asset Allocation for Robo-Advisors

Table 4: Linear dependence between the four assets (hedging portfolios)
Asset
1
2
3
4

αi
1.70%
2.06%
2.85%
1.41%

βi
0.139
0.230
0.409
0.750

0.354
0.347

0.187
0.268

0.250
0.191
0.045

0.063

R2i
45.83%
37.77%
33.52%
41.50%

Table 5: Risk/return analysis of hedging portfolios
Asset
1
2
3
4

µi
7.00%
8.00%
9.00%
10.00%

µ̂i
5.30%
5.94%
6.15%
8.59%

αi
1.70%
2.06%
2.85%
1.41%

σi
15.00%
18.00%
20.00%
25.00%

σ̂i
10.16%
11.06%
11.58%
16.11%

si
11.04%
14.20%
16.31%
19.12%

R2i
45.83%
37.77%
33.52%
41.50%

Table 6: Optimal portfolio
Asset
1
2
3
4

ωi
84.62%
60.68%
50.43%
70.94%

yi?
80.22%
63.67%
58.02%
41.26%

zi?
132.48%
125.09%
118.19%
85.40%

x?i
36.00%
26.39%
27.67%
9.94%

We then calculate the risk/return statistics of hedging portfolios in Table 5. We verify
that the following equalities hold7 : µi = µ̂i + αi and σi2 = σ̂i2 + s2i . Finally, we obtain the
optimal portfolio given in Table 6. γ is set to 0.2578 in order to obtain a 100% exposure.
In this example, the optimal portfolio is: x?1 = 36%, x?2 = 26.39%, x?3 = 27.67% and
x?4 = 9.94%. There is no short position, because the alpha αi is positive for all the assets,
meaning that hedging portfolios are not able to produce a better expected return than the
corresponding assets.
We now modify the correlation between the third and fourth assets, and set ρ3,4 = 95%.
This high correlation changes the results of the linear regression (see Tables 7 and 8). Indeed,
the coefficient of determination for Assets 3 and 4 is larger than 90%, and the fourth hedging
portfolio has an expected return that is higher than that of the fourth asset. Since α4 is the
only negative alpha, the optimal portfolio is short on the fourth asset and long on the other
assets (see Table 9). Another important factor is the impact of R2i on the weights ωi . Thus,
ω3 and ω4 are larger than 10 whereas ω1 and ω2 are smaller than 1. Even if the difference
between yi? and zi? is the smallest for Assets 3 and 4, the leverage effect largely compensates
the long/short effect, and explains why the optimal portfolio has a large exposure on Assets
3 and 4.
The theoretical analysis presented in this paragraph also highlights the importance of the
expected returns. Indeed, even if they do not change the composition and the risk analysis
7 We have:

and:

h
i
h
i
(−i)
µ̂i = E R̂i,t = E βi> Rt
= βi> µ(−i)




(−i)
σ̂i2 = var R̂i,t = var βi> Rt
= σi2 R2i

9

Robust Asset Allocation for Robo-Advisors

Table 7: Linear dependence between the four assets (ρ3,4 = 95%)
Asset
1
2
3
4

αi
3.16%
2.23%
1.66%
−1.61%

βi
0.244
0.443
−0.174
0.292

−0.595
0.470

0.076
−0.035

0.724
−0.157
0.795

1.094

R2i
47.41%
33.70%
91.34%
92.37%

Table 8: Risk/return analysis of hedging portfolios (ρ3,4 = 95%)
Asset
1
2
3
4

µi
7.00%
8.00%
9.00%
10.00%

µ̂i
3.84%
5.77%
7.34%
11.61%

αi
3.16%
2.23%
1.66%
−1.61%

σi
15.00%
18.00%
20.00%
25.00%

σ̂i
10.33%
10.45%
19.11%
24.03%

si
10.88%
14.66%
5.89%
6.90%

R2i
47.41%
33.70%
91.34%
92.37%

Table 9: Optimal portfolio (ρ3,4 = 95%)
Asset
1
2
3
4

ωi
90.16%
50.82%
1054.10%
1211.48%

yi?
60.73%
48.20%
43.92%
31.23%

zi?
70.30%
103.08%
39.22%
39.25%

x?i
52.10%
20.31%
93.44%
−65.85%

of hedging portfolios, they impact the return analysis. An example is provided in Appendix
C.1 on page 63. We change the expected return of the first asset and set µ1 = 3%. In this
case, the expected return of the first asset is largely smaller than the expected return of
the corresponding hedging portfolio. At the same time, the alpha of the other three assets
increases sharply. This is why Markowitz optimization increases the allocation in the third
asset and takes a short position on the first asset.
Let us write Equation (8) as follows:


X
Ri,t − µi
Rj,t − µj
=
β̃i,j
+ εi,t
σi
σj
j6=i

where the coefficients β̃i,j only depend on the correlation matrix C. We have the following
correspondence:
 
X
µj
αi = µi − σi
β̃i,j
σj
j6=i

and:


βi,j = β̃i,j

σi
σj



Moreover, we notice that:
s2i = σi2 1 − R2i
and:





−1 

(−i)
R2i = e>
C (−i)
C (−i) ei
i C

where C (−i) is the correlation matrix excluding the ith asset. We obtain the following effects:
10

Robust Asset Allocation for Robo-Advisors

• A change in the expected return µi impacts the alpha αi of the hedging portfolios. It
does not change the composition βi of hedging portfolios or the weights ωi ;
• A change in the volatility σi impacts the exposures βi of the hedging portfolios. It does
not change the weights ωi , but modifies the value of alphas. As such, the composition
of the portfolio zi changes;
• A change in the correlation ρi,j impacts all the parameters (αi , βi and wi ).
We also notice that the correlations are the only parameters that are used for calculating
the coefficient of determination R2i . Therefore, correlations are the key parameters for
understanding the leverage effects in the Markowitz model. Indeed, they impact both the
tracking error volatilities si and the weights ωi . The main effect of the volatility σi concerns
the tracking error, because si is an increasing function of σi . A high volatility σi therefore
negatively impacts the allocation yi and zi .

3

Theory of regularization

The stability issue has been considered by Michaud (1989) in a very famous publication “The
Makowitz Optimization Enigma: Is Optimized Optimal?”. In his works, Michaud clearly
makes the distinction between mathematical optimization and financial optimality. For
instance, if we consider two assets that are highly similar in terms of risk and return, a fund
manager will most likely spread a long exposure into these two assets, whereas Markowitz
will play an arbitrage between them. Academics have proposed several approaches to make
Markowitz’s solutions more robust. Two main directions have been explored. The first one
concerns the regularization of the covariance matrix. As seen in Equation (7), the problem
is ill-conditioned because of the magnitude of eigenvectors. One solution is therefore to
change the eigenvalues of Σ. For instance, the direct approach consists in deleting the lowest
eigenvalues (Laloux et al., 1999). The indirect approach mixes different covariance matrices
in order to obtain a more robust estimator, and is called the shrinkage method (Ledoit and
Wolf, 2003). The second direction concerns the regularization of the optimization problem
(e.g. adding L2 penalty) or the sparsity of the solution (e.g. adding L1 penalty). The
simplest way is to add some weight constraints. For instance, we can impose that the
sum of weights is equal to one, the weights are positive, etc. Another approach consists in
modifying the objective function by adding some penalties, such as ridge or lasso norms.

3.1

Adding constraints

Let us specify the Markowitz problem in the following way:
min
s.t.

1 >
x Σx
2

 1> x = 1
x> µ ⩾ µ?

x∈Ω

where Ω is the set of weight constraints. This is a variant of the µ-problem (2) described on
page 3. We consider two optimized portfolios:
• The first one is the unconstrained portfolio x? (µ, Σ) with Ω = Rn .
• The second one is the constrained portfolio x̃ (µ, Σ) with some constraints added.
11

Robust Asset Allocation for Robo-Advisors

Jagannathan and Ma (2003) assume that the weight of asset i is between a lower bound x−
i
and an upper bound x+
i :

+
Ω = x ∈ Rn : x −
i ⩽ xi ≤ xi
They show that the constrained optimal portfolio is the solution of the unconstrained problem:


x̃ (µ, Σ) = x? µ̃, Σ̃
with:


µ̃ = µ
>
Σ̃ = Σ + (λ+ − λ− ) 1> + 1 (λ+ − λ− )

where λ− and λ+ are the Lagrange coefficients vectors associated with the lower and upper
bounds. Introducing weight constraints is then equivalent to using another covariance matrix
Σ̃, or shrinking the covariance matrix. More generally, if we introduce linear inequality
constraints:
Ω = {x ∈ Rn : Cx ⩾ d}
we obtain a similar result. The covariance matrix is shrunk as follows8 :
Σ̃ = Σ − C > λ1> + 1λ> C



where λ is the vector of Lagrange coefficients associated with the constraints Cx ⩾ d.
We again consider the previous example given on page 5. If we compute the global
minimum variable, the solution x? is equal to 65.57%, 29.06%, 13.61% and −8.24%. Let us
suppose that the portfolio manager is not satisfied with this optimized portfolio and decides
to impose some constraints. For instance, he could decide that the portfolio must contain at
least 10% of all assets. In order to achieve a certain degree of diversification, he could also
+
decide to impose an upper bound of 40%. With these constraints x−
i = 10% and xi = 40%,
the solution becomes 40.00%, 31.18%, 18.82% and 10.00%. Thanks to the Jagannathan-Ma
framework, we can compute the shrinkage covariance matrix9 , and deduce the shrinkage
volatilities σ̃i and correlation matrix ˜
C, which are reported in Table 10. To obtain this
new solution, one must increase (implicitly) the volatility of the first asset, and decrease
(implicitly) the volatility of the fourth asset. Concerning the correlations, we also notice
that they have changed. In Table 11, we report the results when the objective function is
to target an expected return of 9%. In this case, we notice that introducing constraints is
equivalent to introducing some views on the first asset. Indeed, this allows us to impose a
better Sharpe ratio and a lower correlation with the second asset.

Table 10: Jagannathan-Ma shrinkage of the GMV portfolio
Asset
1
2
3
4

x?i
65.57%
29.06%
13.61%
−8.24%

x̃i
40.00%
31.18%
18.82%
10.00%

σ̃i
16.80%
18.00%
20.00%
22.96%

C˜
100.00%
54.10%
53.16%
53.07%

100.00%
50.00%
42.61%

100.00%
32.90%

100.00%

8 The shrinkage covariance matrix is not necessarily positive definite (Roncalli, 2013).
9 We have λ− = 48.89 bps and λ+ = 28.58 bps. The other Lagrange coefficients are equal to zero.
4
1

12

Robust Asset Allocation for Robo-Advisors

Table 11: Jagannathan-Ma shrinkage of the MVO portfolio (µ? = 9%)
Asset
1
2
3
4

x?i
3.30%
23.44%
43.21%
30.05%

x̃i
10.00%
15.00%
40.00%
35.00%

σ̃i
12.06%
18.00%
20.59%
25.00%

C˜
100.00%
43.87%
49.20%
61.43%

100.00%
51.79%
50.00%

100.00%
41.18%

100.00%

Remark 2 Constraints are inherent to Markowitz optimization. Indeed, the raw solution
given by the mean-variance optimization is generally not satisfied. This is why Quants spend
a lot of time adding and testing constraints. This is particular true for strategic asset allocation, for which the annual exercises are very time-consuming. However, adding constraints
introduces the personal views of the Quant in charge of the optimization. Moreover, this
process of trial and error must be repeated each time the allocation problem changes. Therefore, Markowitz optimization is more a handmade solution, and not an industrial solution.
This is why it cannot be used “as is” by robo-advisors, whose mass production/customization
approach is incompatible with human intervention.

3.2

Adding a benchmark

Let us now consider a benchmark which is represented by a portfolio b. The tracking error
between the portfolio x and its benchmark b is the difference between the return of the
portfolio and the return of the benchmark:
et

= Rt (x) − Rt (b)
=

>

(x − b) Rt

where Rt = (Rt,1 , . . . , Rt,n ) is the vector of asset returns. The expected excess return is:
>

µ (x | b) = E [et ] = (x − b) µ
whereas the volatility of the tracking error is:
q
>
σ (x | b) = σ (et ) = (x − b) Σ (x − b)
The investor’s objective is to maximize the expected tracking error with a constraint on the
tracking error volatility. Like the Markowitz problem, we transform this σ-problem into a
γ-problem:
x? (γ)

=

arg min

s.t.

x∈Ω

1
>
(x − b) Σ (x − b) − γx> µ (x | b)
2

The objective function is then:
f (x)

=
=

1
>
>
(x − b) Σ (x − b) − γ (x − b) µ
2


1 >
1 >
>
>
x Σx − x (γµ + Σb) +
b Σb + γb µ
2
2
13

Robust Asset Allocation for Robo-Advisors

We deduce that:
x? (γ)

1 >
x Σx − γx> µ̃
2
s.t. x ∈ Ω
=

1
Σb. Let µb be the vector of implied expected returns such that the
γ
benchmark b is the optimal portfolio. Since we have b = γΣ−1 µb , the optimization problem
becomes:


µ + µb
1 >
>
?
x (γ) = arg min x Σx − ξx
2
2
s.t. x ∈ Ω
where µ̃ = µ +

where ξ = 2γ. Introducing a benchmark constraint is then equivalent to regularizing the
expected returns.

3.3

Tikhonov and ridge regularization

Previously, we have seen a method that regularizes the covariance matrix and an approach
that regularizes the vector of expected returns. We now turn to a framework that regularizes
the two input parameters of Markowitz optimization problems, and not only the covariance
matrix or the vector of expected returns. While the two previous approaches are more
specific to financial optimization, the following methods have been developed in PDEs and
later in statistics. This is why we consider the following general optimization problem:
x?

=
s.t.

1
2
arg min kA1 x − b1 k2
2

A2 x = b2
A3 x ⩾ b3

(9)

We recognize a standard quadratic programming problem. Problems (1) – (6) can easily be
written as Problem (9). For instance, the γ-problem (3) is obtained with A>
1 A1 = Σ and
>
A>
and b3 = µ? for the µ-problem. If we prefer
1 b1 = γµ, while we have b1 = 0, A3 = µ
1/2
to use the empirical model (6), we specify A1 = W 1/2 R = Dw CT R and b1 = γW −1/2 w =

−1
1/2
γ CT> Dw
w. We notice that the L2 norm is natural because of the specification of A1 .
3.3.1

Formulation of the Tikhonov problem

In order to regularize the Markowitz optimization problem, we can add a penalty term. For
instance, the most famous approach is the Tikhonov regularization. The general problem
can be written as follows:
x?

1
1
2
2
kA1 x − b1 k2 + %2 kΓ2 (x − x0 )k2
2
2
s.t. A2 x = b2
=

arg min

(10)

where %2 > 0 is a positive number, Γ2 ∈ Rn×n , A2 ∈ Rm×n and b2 ∈ Rm×1 . The vector
x0 is an initial solution. The Tikhonov regularization matrix Γ2 forces the solution to be
close to x0 with respect to the semi-norm x 7→ kΓ2 xk2 whereas the Tikhonov regularization
parameter %2 indicates the strength of the regularization.
14

Robust Asset Allocation for Robo-Advisors

Remark 3 In portfolio optimization, x0 can be seen as a reference portfolio. For instance,
it can be a benchmark, an heuristic portfolio10 or the investment portfolio of the previous
period. The L2 penalty term may then be used to control the deviation between the new
portfolio and the reference portfolio, the tracking error or the portfolio turnover.
Remark 4 The previous approach was introduced in asset management by Jorion (1988,
1992), who considered the Bayes-Stein estimator based on the one-factor model developed by
Sharpe (1963). With the notations above, we have the following correspondence: Γ2 = 11>
and x0 = 0.
In Appendix A.4 on page 48, we show that the optimal solution is the x-coordinate of
the linear system solution11 :
 >

  >

>
x
A1 A1 + %2 Γ>
A1 b1 + %2 Γ>
2 Γ 2 A2
2 Γ2 x0
=
(11)
λ
A2
0
b2
where λ is the vector of Lagrange coefficients associated with the constraint A2 x = b2 . The
OLS regression corresponds to Γ2 = 0 whereas the ridge regression is obtained with Γ2 = In .
−1 >
For λ = 0 and %2 = 0, the OLS solution is simply x? = A†1 b1 where A†1 = A>
A1
1 A1
is the Moore-Penrose pseudo-inverse matrix of A1 . For λ = 0 and %2 > 0, the regularized
#
#
solution becomes x? = A#
1 b1 where A1 may be interpreted as the Tikhonov regularization
of A†1 :
−1 >
>
>
A#
A1
1 = A1 A1 + % 2 Γ 2 Γ 2
>
>
We also notice that
 A1 A1 + %2 Γ2 Γ2 is invertible if the matrix Γ2 is invertible. Indeed, if
>
>
A1 A1 + %2 Γ2 Γ2 x = 0, we have:

2
2
2
>
0 = x > A>
1 A1 + %2 Γ2 Γ2 x = kA1 xk2 + %2 kΓ2 xk2 ⩾ %2 kΓ2 xk2
>
This ensures the property that the matrix A>
1 A1 + %2 Γ2 Γ2 is positive definite. This idea
can be extended using spectral decomposition of A1 , which naturally leads to defining the
regularization of the matrix A1 through spectral filters.

3.3.2

Relationship with covariance shrinkage methods

Let us consider the regularized Markowitz problem:
x?

1
arg min x> Σx − γx> µ + R (x)
2
>
s.t. 1 x = 1
=

where R (x) is the regularization function. If we consider the Tikhonov formulation (10),
>
we have the following correspondence: A>
1 A1 = Σ and A1 b1 = γµ. We deduce that the
†
regularization on the matrix A1 can be written as a regularization on the covariance matrix
Σ when there is no target portfolio (x0 = 0):
Σ (%2 ) = Σ + %2 Γ>
2 Γ2
Therefore, there is a strong relationship between regularization and shrinkage. Indeed, the
empirical covariance matrix Σ̂ is an unbiased estimator of Σ, but its convergence is very slow
10 For instance, it can be the equally-weighted (EW) portfolio or the equal risk contribution (ERC) portfolio
(Roncalli, 2013).
11 We obtain a linear system of the form Az = b where A is a symmetric 2 × 2 block matrix. The (1,1)
block depends on the matrix A1 while the (2,1) block depends on the matrix A2 .

15

Robust Asset Allocation for Robo-Advisors

in particular when n is large. We know also that the estimator Φ̂ based on factor models
converges more quickly, but it is biased. Ledoit and Wolf (2003) propose combining the two
estimators Σ̂ and Φ̂ in order to obtain a more efficient estimator. Let Σ̂ (α) = αΣ̂+(1 − α) Φ̂
be this new estimator. Ledoit and Wolf estimate the optimal value of α by minimizing the
expected value of the quadratic loss:
α? = arg min E [L (α)]
where the loss function is equal to:
2

L (α) = αΣ̂ + (1 − α) Φ̂ − Σ
2

We have, up to a scaling factor12 , the following correspondence:

1 − α?

%2 =
α?
 Γ = chol Φ̂
2

where chol M is the upper Cholesky factor of the matrix M . Therefore, the Ledoit-Wolf
shrinkage technique is a special case of Tikhonov regularization. In a similar way, the double
shrinkage method proposed by Candelon et al. (2012) is obtained by setting Γ2 = In and
x0 6= 0.
3.3.3

Ridge regularization

The ridge regularization is defined by Γ2 = In . We deduce that the mean-variance objective
function becomes:
f (x)

=
∝
=

1 >
1
2
x Σx − γx> µ + %2 kx − x0 k2
2
2
1 >
x (Σ + %2 In ) x − x> (γµ + %2 x0 )
2
1 >
x Σ (%2 ) x − γx> µ (%2 )
2
%2
x0 . Let x? (γ; %2 , x0 ) be the unconstrained
γ

where Σ (%2 ) = Σ + %2 In and µ (%2 ) = µ +
solution of the ridge optimization problem:

1
1
2
x? (γ; %2 , x0 ) = arg min x> Σx − γx> µ + %2 kx − x0 k2
2
2
We have:
x? (γ; %2 , x0 )

−1

= γΣ (%2 )

µ (%2 )


%2
−1
= γ (Σ + %2 In )
µ + x0
γ

−1 −1
?
= In + %2 Σ
(x (γ; µ) + x? (%2 ; x0 ))

where x? (γ; µ) = γΣ−1 µ is the Markowitz solution. We deduce that the regularized solution
is the average of two portfolios: the Markowitz portfolio x? (γ; µ) and the optimal portfolio
12 This is not an issue since γ is not a fixed parameter, but is calibrated to solve a σ-problem or a µ-problem.

16

Robust Asset Allocation for Robo-Advisors

x? (%2 ; x0 ) when the vector of expected returns is equal to x0 and the risk/return trade-off
parameter is %2 . Bruder et al. (2013) also show that:
x? (γ; %2 , x0 ) = ω (%2 ) x? (γ; µ) + (In − ω (%2 )) x0
where the matrix of weights ω (%2 ) is equal to In + %2 Σ−1

−1

. We verify that:

lim ω (%2 ) = 0

%2 →∞

Without any constraints, the ridge regularization reduces the leverage of Markowitz portfolio
when there is no target portfolio. When we impose that the portfolio is fully invested
(1> x = 1), this is equivalent imposing that the target portfolio is the equally-weighted
portfolio.
We consider an example where the investment universe is composed of 4 assets. The
expected returns are equal to µ1 = 4%, µ2 = 5%, µ3 = 9% and µ4 = 10% whereas the
volatilities are equal to σ1 = 15%, σ2 = 18%, σ3 = 20% and σ4 = 25%. The correlation
matrix is the following:


1.00

 0.70
1.00

C=

 0.10
0.10
1.00
−0.20 −0.20 −0.70 1.00
We assume that γ = 0.25 and the portfolio is fully invested. We impose that the target
portfolio x0 is equal to (40%, 30%, 20%, 10%). Figure 1 show the optimal weights with
respect to the penalization factor %2 . We verify that the optimized portfolio converges
to the target portfolio when %2 increases. When there is no target portfolio, it converges
to the equally-weighted portfolio (see Figure 2). This result
Pn is due to the capital budget
constraint. Indeed, if we do not impose the constraint
i=1 xi = 1, the ridge portfolio
converges to the zero solution x? = 0. We also notice that the paths of weights are not
necessarily monotonous (increasing or decreasing). For instance, the weight of the second
asset decreases when %2 is small and increases when %2 is large.
We notice that the ridge regularization
impacts entirely the covariance matrix. Indeed,
p
the shrinkage volatilities are equal to σi2 + %2 whereas the shrinkage correlation matrix is
defined by:
σi σj
[C (%2 )]i,j = ρi,j q

2
(σi + %2 ) σj2 + %2
It follows that lim%2 →∞ C (%2 ) = In . Since the volatilities tend to ∞, the ridge regularization
can be viewed as a shrinkage covariance method between the input covariance matrix Σ and
the identity matrix:
Σ (α) = αΣ + (1 − α) In
Remark 5 A variant of the ridge regularization is to define Γ2 as a diagonal matrix. For
instance, if Γ2 = diag Σ, the regularized correlation matrix satisfies:
[C (%2 )]i,j =

ρi,j
1 + %2

In Figure 3, we have reported the impact of the parameter %2 on the correlation values.

17

Robust Asset Allocation for Robo-Advisors

Figure 1: Ridge regularization with a target portfolio
60

45

30

15

0

-15

-30
10-4

10-3

10-2

10-1

100

101

Figure 2: Ridge regularization without a target portfolio
60

45

30

15

0

-15

-30
10-4

10-3

10-2

10-1

18

100

101

Robust Asset Allocation for Robo-Advisors

Figure 3: Impact of the parameter % on the correlation (diagonal ridge regularization)
100
75
50
25

-100

-75

-50

-25

25

50

75

100

-25
-50
-75
-100

3.4

Spectral filtering

Spectral filtering is a general approach based on the singular value decomposition (SVD) of
the matrix A1 . Ridge regularization and denoising techniques can be seen as special cases
of the SVD method.
3.4.1

General filters

We consider the SVD decomposition of the matrix A1 by assuming that rank A1 = r:
A1 = U SV >
where the matrices13 U ∈ Rn×r , V ∈ Rn×r , s = (s1 , . . . , sr ) ∈ Rr and S = diag (s) satisfy
U > U = V > V = Ir and sk ⩾ sk+1 > 0. The Moore-Penrose pseudo-inverse of A1 can be
defined as:
A†1 = V S −1 U >

where S −1 = diag s† . Let us denote smax (A1 ) = s1 the largest singular value of A1 .
As instability is raised by small eigenvalues, filtering can be applied to keep eigenvalues
away from 0. A filter G (s; %) = (G (s1 ; %) , . . . , G (sr ; %)) is a vector-valued function, where
the k th entry G (sk ; %) : ]0, smax (A)] → R satisfies:
lim G (sk ; %) =

%→0

1
sk

for all % ⩾ 0 and sk ∈ ]0, smax (A)]. The parameter % controls the magnitude of the regularization of A†1 :
A†1 (%) = V diag (G (s; %)) U >
13 In the case of the empirical model, we have U ∈ RT ×r .

19

Robust Asset Allocation for Robo-Advisors

As a consequence, we verify the property of convergence:
lim A†1 (%) = A†1

%→0

This method can be extended to regularize the matrix A>
A . On one hand, if A1 has

−1 1 1
†
>
full rank, we can approximate Q = A>
. On the other hand, a direct
1 A1 by A1 A1 (%)
2 >
>
computation leads to Q = A>
1 A1 = V S V . Therefore, we can regularize Q = A1 A1 by:

Q (%) = V diag s2 (%) V >
†

†

where s2 (%) is a vector that may be equal to G (s; %)
s. Once again, we have the convergence property:

G (s; %) , or G (s

†

†

s; %) or G (s; %)

lim Q (%) = A>
1 A1

%→0

If we consider the problem:
x?

=
s.t.

1
2
kA1 x − b1 k2
2
A2 x = b2
arg min

the normal equations are:


A>
1 A1
A2

A>
2
0



x
λ




=

A>
1 b1
b2


(12)

Spectral filtering is then equivalent to replacing the linear system (12) by the following set
of normal equations:


!




†
x
V diag s2 (%) V > A>
U diag G (s; %) V > b1
2
=
(13)
λ
A2
0
b2
3.4.2

Application to Tikhonov regularization

To define the spectral regularization of the Tikhonov problem, the matrices A1 and Γ2 have
to be able to be factored in a coherent way:
A1 = U S 1 V >
and:
Γ2 = W S2 V >
Direct computations gives:
 >
>
2
2
A>
1 A1 + %2 Γ2 Γ2 = V S1 + %2 S2 V
We deduce that the k th entry of the spectral filter G (s1 ; %2 ) is defined by:
G (s1,k ; %2 ) =

s1,k
2
s1,k + %2 s22,k

20

Robust Asset Allocation for Robo-Advisors

Using the previous notations, we have:
%2 Γ>
2 Γ2

>
>
A>
1 A1 + %2 Γ2 Γ2 − A1 A1

= V diag s21 (%2 ) V > − V diag (s1

= V diag s21 (%2 ) − s21 V >

=

s1 ) V >

where s21 = s1 s1 . In this case, the optimal portfolio x? is the x-coordinate of the solution
to the linear system:




  >

x
V diag s21 (%2 ) V > A>
A1 b1 + V diag s21 (%2 ) − s21 V > x0
2
=
(14)
λ
A2
0
b2
We notice that only the right singular vectors appear in Equation (14). Ridge regularization
can be viewed as particular filters14 . More generally, when A1 and Γ2 have the same right
singular vectors, Tikhonov regularization can be stated in terms of a filter.
In Figure 4, we report the spectral filter of the ridge regularization. The spectral filtering
approach includes another popular method, which is the denoising method (Laloux et al.,
1999):
G (s1,k ; %2 ) = 1 {|s1,k | ⩾ %2 } · s†1,k
We notice that deleting singular values is equivalent to applying a hard thresholding method
while ridge regularization is a smoothing approach.
3.4.3

Improvement of the stability condition

The condition number κ (A) of the matrix A summarizes the level of difficulty when performing the optimization in a stable way. More specifically, it measures how much an error
on the vector b changes the solution of the linear equation Ax = b. We have:
κ (A) = A† · kAk

†

It follows that κ A = κ (A), and we have the property κ (A) ⩾ 1. When κ (A) is low, the
problem is numerically stable and easy to solve. The closer to one, the better the stability.
With the L∞ norm, we obtain:
κ (A) =

maxk |sk |
mink |sk |

(15)

where the sk ’s are the singular values of A. Using the filter G (s; %), we obtain:

mink |G (sk ; %)|
κ A† (%) =
maxk |G (sk ; %)|

(16)

For a fixed value of % > 0, all previous filters satisfy the two following properties:
1. G (sk ; %) ∼ s−1
k for sk → ∞;
2. G (sk ; %) is bounded from above on [0, +∞).
As a consequence, if we compare Equations (15) and (16), the denominator is essentially
unchanged while the numerator is decreased15 . Therefore, spectral filtering decreases the
condition number of A, because these techniques reduce the dispersion of singular values.

s1,k ; %2 = s†1,k . For Γ2 = In (ridge regularization), the kth entry of the spectral
filter G (s1 ; %2 ) is defined by:

s1,k
G s1,k ; %2 = 2
s1,k + %2
14 For Γ

2 = 0, we have G

15 From an unbounded function to a bounded function.

21

Robust Asset Allocation for Robo-Advisors

Figure 4: Spectral filtering (ridge regularization and denoising method)

1.5

1

0.5

-5

-4

-3

-2

-1

1

2

3

4

5

-0.5

-1

-1.5

3.5

Mixed penalties

The Euclidian regularization is natural because the L2 norm appears in Problem (3). Explicit
formulas are obtained, and can be implemented at once. Other regularization techniques
have been introduced to impose other constraints on the optimal solution x? . As the unit
ball for the L1 norm is not uniformly convex, sparse solutions may be obtained by penalizing
with L1 instead of L2 .
3.5.1

Lp regularization

Instead of Tikhonov regularization, one may consider the Lp regularization:
x?

=
s.t.

1
1
2
p
kA1 x − b1 k2 + %p kΓp (x − x0 )kp
2
p
A2 x = b2
arg min

(17)

where x0 ∈ Rn is a targeted portfolio and p > 0.
p
For p > 1, the function Γp (x) = kΓp (x − x0 )kp is strictly convex and its gradient is
p−1

Lipschitz continuous. Indeed, the gradient is equal pΓ>
,
p sign (Γp (x − x0 )) |Γp (x − x0 )|
where the functions sign (x) and |x| are taken component wise. For p = 1, the function
Γ1 (x) is convex, lower semi-continuous but may not be differentiable at x = x0 . An explicit
expression for its subgradient can be formulated in terms of proximal operators. For p ∈
]0, 1[, the function Γp (x) is not convex, and Problem (17) is not convex.
The penalties Lp for p ⩾ 1 are used for regularization, while the penalties Lp for p ⩽ 1
are used for sparsity. The case p = 1 is the most interesting since it corresponds to the lasso
regression (Tibshirani, 1996). In this case, a large value of %1 associated with the constraint
1> x = 1 forces the optimal portfolio to have long-only positions (Brodie et al., 2009).
22

Robust Asset Allocation for Robo-Advisors

Figure 5: Lasso regularization with a target portfolio
60

45

30

15

0

-15

-30
0

0.5

1

1.5

2

2.5

Figure 6: Lasso regularization without a target portfolio
60

45

30

15

0

-15

-30
0

0.05

0.1

0.15

23

0.2

0.25

0.3

Robust Asset Allocation for Robo-Advisors

We consider the example given on page 17. We use a L1 (or lasso) penalty with Γ1 = In .
Figure 5 show the optimal weights with respect to the penalization factor %1 . Like in the ridge
approach, the optimized portfolio converges to the optimal portfolio when the parameter %
increases. When there is no target portfolio, we observe a divergence of the limit portfolio
between ridge and lasso approaches. While the ridge portfolio converges to the equallyweighted portfolio, the lasso portfolio converges to the long-only mean-variance optimized
portfolio (Figure 6). If we compare Figures 1 and 5, we notice that the magnitude of the
regularization factor is not the same. We also observe that the paths are different. The
path is smoothed and continuous for the ridge approach, while it is more a piecewise linear
function for the lasso approach. We verify that the L1 penalty produces a sparse optimized
portfolio. This is obvious for the case where there is no target portfolio since weights may be
equal to zero. When there is a target portfolio, the sparsity concerns the bets between the
optimized portfolio x? and the target portfolio x0 . In this case, relative (and not absolute)
weights are equal to zero. Another difference between the two approaches is that the lasso
method produces a monotonic path (decreasing or increasing) contrary to the ridge method.
3.5.2

L1 − L2 regularization

We can also consider a mixed penalty:
x?

=
s.t.

1
1
2
p
2
kA1 x − b1 k2 + %p kΓp (x − x0 )kp + %2 kΓ2 (x − x0 )k2
2
2
A2 x = b2
arg min

(18)

where p 6= 2. In the case p = 1, we obtain:
x?

=
s.t.

1
1
2
2
kA1 x − b1 k2 + %1 kΓ1 (x − x0 )k1 + %2 kΓ2 (x − x0 )k2
2
2
A2 x = b2
arg min

(19)

This regularization is called elastic net (Hastie et al., 2009). This is the most common mixed
penalty used in portfolio optimization (Roncalli, 2013).
We consider again the example given on page 17. We use a lasso-ridge penalty with
Γ1 = Γ2 = In . Results are reported in Figures 7 and 8. We notice a large difference
concerning the convergence. Indeed, we recall that the lasso and ridge approaches converge
to the same portfolio when we impose a target portfolio, but to two different portfolios when
there is no target portfolio. When mixing the two norms, the limit portfolio is generally
the ridge portfolio, because of the magnitude of %1 and %2 in portfolio management (see
Appendix A.5 on page 49). This result is true because we have imposed Γ1 = Γ2 = In .
3.5.3

Solving the mixed penalty problem

Problems (18) and (19) are more complex to solve than a traditional quadratic programming
problem. In the case of the L1 − L2 regularization problem and if we assume that Γ1 is a
matrix with non-negative entries16 , we can use a modified QP solver. The underlying idea
is to write Γ1 (x) in the following way:
Γ1 (x) = 1> Γ1 δ − + 1> Γ1 δ +
where δ − = max (0, x0 − x) and δ + = max (0, x − x0 ). Therefore we obtain a standard
QP problem by augmenting the vector of unknown variables17 . Thus, the optimization is
16 Which is generally the case (Bruder et al., 2013; Roncalli, 2013).
17 See Appendix A.6 on page 49 for a comprehensive presentation.

24

Robust Asset Allocation for Robo-Advisors

Figure 7: Mixed regularization with a target portfolio

40

30

20

20

0
10

-20
0

0

2

-2
-4

2

-2

1
0

1
-4

0

-4

0

60
40
40
20
20
0

0

2

-2
-4

2

-2

1
0

1

Figure 8: Mixed regularization without a target portfolio

20

20

0
10
-20
0
0

0

2

-2
-4

2

-2

1
0

1
-4

0

-4

0

50
60
40
40
30
0

0

2

-2
-4

2

-2

1
0

25

1

Robust Asset Allocation for Robo-Advisors

performed with respect to y = (x, δ − , δ + ) and no longer with respect to x. In the other
cases, when we consider an Lp penalty with p 6= 2 or when Γ1 is a matrix with some negative
entries, the general approach is to use the ADMM algorithm, which is described in Appendix
A.7 on page 50. For instance, Problem (19) can be written as:
{x? , z ? }

where:
f (x) =

=

arg min f (x) + g (z)

s.t.

x−z =0

1
1
2
2
kA1 x − b1 k2 + %2 kΓ2 (x − x0 )k2 + 1Ω (x)
2
2

and:
g (z)

=

%1 Γ1 (x)

=

%1 kΓ1 (x − x0 )k1

where Ω = {x ∈ Rn : A2 x = b2 }. The interest of this choice is that the x-step includes
the constraint and can be explicitly computed18 , while the z-step requires to compute the
proximal operator of the function Γ1 (x):


2
ϕ (k+1)
z (k+1) = arg min g (z) +
x
− z + u(k)
2
2
The update of the scaled dual variable is:


u(k+1) = u(k) + x(k+1) − z (k+1)
The previous results can be extended when p 6= 1 and Ω is a set of more complex constraints.
Appendices A.7 and A.8 on pages 50–59 contain all the information for solving the
following optimization problem:
x?

=

arg min

s.t.

x∈Ω

1
1
2
p
2
kA1 x − b1 k2 + %p kΓp (x − x0 )kp + %2 kΓ2 (x − x0 )k2
2
2

where Ω may be equality, inequality, bound and Lq norm constraints.

3.6

Optimal choice of the regularization factor

To choose the optimal regularization parameter, we first have to define an optimization
criterion. For instance, the optimal value of %1 or %2 is generally obtained by cross-validation
techniques. Exhaustive methods such as leave-p-out cross-validation (LpOCV) or leaveone-out cross-validation (LOOCV) are computationally intensive. This is why it may be
better to use non-exhaustive methods such as k-fold cross-validation or out-of-sample testing.
However, in the case of the Tikhonov regularization, an explicit formula is known. Indeed,
the generalized cross-validation procedure for choosing %2 does not depend on the dual
variable or the constraints. In the case of the L1 penalty, no explicit formula is known and
the brute force algorithm must be used for finding the optimal value of %1 .
18 We have



2
ϕ
x + z (k) u(k)
x(k+1) = arg min f (x) +
2
2

26

Robust Asset Allocation for Robo-Advisors

3.6.1

Cross-validation and the PRESS statistic

>
T ×K
Let us consider the data matrix X = x>
where xt ∈ RK , and a response
1 , . . . , xT ∈ R
vector Y = (y1 , . . . , yT ) ∈ RT where yt ∈ R. Since the Tikhonov regularization problem is
defined as follows:
1
1
2
2
β̂ = arg min kY − Xβk2 + %2 kΓ2 βk2
2
2
we have:
β̂ = S (%2 ) X > Y
where:
S (%2 ) = X > X + %2 Γ2 Γ>
2

−1

It follows that β̂ is a function of %2 . Therefore, the underlying idea is to find the optimal
value %̂2 .
In order to accurately estimate the hyperparameters of the model and to avoid overfitting
problems, the cross-validation (CV) method comprises several steps:
1. the sample of data is partitioned into two sets, the training set and the test (or validation) set;
2. the model is fitted on the training set;
3. the model is tested on the validation set.
In order to reduce variability, steps 2 and 3 are performed using different partitions of
the data sample (step 1). The validation results are combined, according to a measure
of fit, to give an estimate of the model predictive performance. The hyperparameters are
then chosen in order to maximize this goodness-of-fit measure. Two types of CV may be
performed: exhaustive and non-exhaustive cross-validation. For the first type, the model is
estimated and tested on all possible ways to divide the original sample into training/test sets.
This type of CV consists of the leave-p-out cross validation (LpOCV). In this approach, p
observations are used in the test set and the remaining observations
are used in the training

set19 . This requires training and validating the model Tp times, which can be extremely
expensive if T is large, even for p = 1. Nevertheless, an explicit expression for the sum of
squares of the errors is known in the case of Tikhonov regression. This formula may lead
to O (T ) operations. For this reason, non-exhaustive cross-validation may be preferred in
practice, such as k-fold CV, holdout method, repeated random sub-sampling, jackknife, etc.
Performing k-fold CV is the most popular tool for model selection (Stone, 1974; Wahba,
1977; Stone, 1978).
In k-fold CV, the sample of data is randomly shuffled and split into k (almost) equally
sized groups, the model is fitted using all but the j th group of data, and the j th group of
data is used for the test set. We repeat the procedure k times, in such a way that each
group is tested exactly once. The k-fold cross validated error is generally computed as:
Ecv =

k
2
1XX
yt − x>
t β (k)
T j=1
t∈Gk

where t ∈ Gk denotes the observations of the k th group and β (k) the estimation of β obtained
by leaving out the k th group. Even in simple cases, it cannot be guaranteed that the function
Ecv has a unique minimum. The simple grid search approach is probably the best approach.
19 The leave-one-out cross validation (LOOCV) procedure corresponds to the special case p = 1.

27

Robust Asset Allocation for Robo-Advisors

The exhaustive Leave-one-out cross validation (LOOCV) is a particular case when k is equal
to the size of the dataset. The LOOCV is asymptotically equivalent to Akaike Information
Criterion (AIC), which is commonly used in statistics (Stone, 1977). Interestingly, For
Tikhonov regression, the cross validated error Ecv has an explicit expression known as the
Predicted Sum of Squares (or PRESS) statistic (Allen, 1971 & 1974).
We note Y−t and X−t the (T − 1) vector and (T − 1) × K matrix by leaving out the tth
observation to the vector Y and the matrix X. We have:
−1 >
>
β̂−t = X−t
Xt + %2 Γ2 Γ>
X−t Y−t
2
The explicit expression for the LOOCV procedure is20 :
Press (%2 )

=

T 
X

yt − x>
t β̂−t

2

t=1

=

T
X

1 − x>
t S (%2 ) xt

−2 

yt − x>
t β̂

2

t=1

=

T
X
t=1

[L (%2 ) Y ]t
[L (%2 )]t,t

!2

where L (%2 ) is the projection matrix defined as:
L (%2 )

= IT − XS (%2 ) X >
= I T − X X > X + %2 Γ2 Γ>
2

−1

X>

If S (%2 ) is a band matrix, which is the case for spline models, the coefficients [L (%2 )]t,t
and [L (%2 ) Y ]t can be computed in O (T ) operations thanks to the Hutchinson-De Hoog
algorithm (Hutchinson and De Hoog, 1985).
3.6.2

GCV for centered data as the selection criterion

The generalized cross-validation (GCV) method is a rotation-invariant version of LOOCV
(Craven and Wahba, 1978). Even if it is not its main purpose, this approach replaces the
factor [L (%2 )]t,t by the average value T −1 trace L (%2 ):
GCV (%2 ) =

T
2
X
T2
>
y
−
x
β̂
t
t
trace2 L (%2 ) t=1

(20)

We deduce that the GCV criterion depends on L (%2 ) and the residual sum of squares
2
PT 
>
y
−
x
β̂
. We recall that H (%2 ) = IT − L (%2 ) is the hat matrix. The value
t
t
t=1
[H (%2 )]t,t is called the leverage value (Craven and Wahba, 1978) and determines the amount
by which the predicted value ŷt = x>
t β̂ is influenced by yt . We also know that trace L (%2 ) =
T − K. From the Woodbury formula, we have21 :
−1 >
L (%2 ) = IT − X X > X + %2 Γ2 Γ>
X
2


−1
−1 >
=
IT + X %2 Γ2 Γ>
X
2
20 Proof is given in Appendix A.9 on page 59.
21 The Woodbury matrix identity is:

(A + BCD)−1 = A−1 − A−1 B C −1 + DA−1 B

28

−1

DA−1

Robust Asset Allocation for Robo-Advisors

Let λt be the eigenvalues22 of the symmetric real matrix X Γ2 Γ>
2
trace L (%2 ) =

T
X
t=1

λt
1+
%2

−1

X > . We have:

−1

−2

This formula allows the value of trace L (%2 ) to be computed for every value of %2 . Like
the PRESS statistic, the optimal value of %2 is obtained by minimizing the GCV function
given by Equation (20).

4

Application to robo-advisory

The previous techniques are of particular interest for portfolio optimization when building
a strategic asset allocation (SAA), a trend-following strategy or more generally a meanvariance diversified portfolio. Depending on the approach, they can diversify or concentrate
the portfolio. By mixing the different approaches, we can also obtain a diversified allocation
on some selected stocks. In this case, portfolio regularization and portfolio sparsity are
combined. The previous techniques can also be used when implementing tactical asset
allocation (TAA). In this case, regularization and sparsity are imposed in a relative way
with respect to a benchmark or a current investment portfolio. In this section, we show
why these techniques are necessary when building a robo-advisor based on an automated
allocation engine.

4.1

Robo-advisory and the secret sauce of portfolio optimization

The idea that portfolio optimization is a simple mathematical problem is mistaken. It is a
process that requires manual interventions and may take considerable time before a solution
is found. And this human intervention has little in common with numerical algorithms.
Indeed, Quants know that the secret sauce of portfolio optimization lies in the alchemy of
defining the right constraints in order to obtain an acceptable solution that makes sense. Let
us consider the traditional strategic asset allocation exercise that is performed by institutional investors almost every year. We assume that the SAA team has already produced the
two inputs: the vector µ of expected returns and the covariance Σ of asset returns. We could
think that the hard work has therefore been done, and that computing the SAA portfolio
will take a matter of seconds since we just have to run a Markowitz optimization. In reality,
solving one Markowitz optimization generally produces a bad solution and is not sufficient.
This is why Quants will use an iterative process based on this optimization program:
x?(k)

=
s.t.

1
arg min x> Σx − γx> µ
2
 >
 1 x=1
0⩽x⩽1

x ∈ Ω(k)

(21)

where Ω(0) = Rn and k is the step. They will begin by solving the traditional Markowitz
problem with long-only constraints and will find an initial solution x?(0) . Then, they will
analyze this solution and define a new set of constraints Ω(1) that might produce a more
acceptable solution. The concept of “acceptable solution” remains unclear, but it means
one that can be accepted by the chief investment officer. Once Ω(1) is defined, Quants
will run the optimization problem (21) and obtain a new solution x?(1) . Next, they will
22 Computing the eigenvalues of X

Γ2 Γ>
2

−1


X > can be done in O T 3 operations.

29

Robust Asset Allocation for Robo-Advisors

analyze this new solution and define a new set of constraints Ω(2) that might produce an
even more acceptable solution. They will iterate this process a number of times. Therefore,
this iterative process can be represented by the sequence P defined as follows:
o
n
P = x?(0) , Ω(1) , x?(1) , Ω(2) , x?(2) , Ω(3) , x?(3) , . . .
Using this tool, we can evaluate Quants and draw some conclusions:
• A good Quant is a person that is able to “close” this sequence in a limited number of
steps.
• A bad Quant is a person that produces an infinite sequence and is not able to end the
process.
• Quant Q1 is more efficient than Quant Q2 if:
card P (Q1 ) < card P (Q2 )
Let us illustrate the previous process with an example23 . We consider a universe of nine
asset classes: (1) US 10Y Bonds, (2) Euro 10Y Bonds, (3) Investment Grade Bonds, (4)
High Yield Bonds, (5) US Equities, (6) Euro Equities, (7) Japan Equities, (8) EM Equities
and (9) Commodities. In Tables 12 and 13, we indicate the statistics used to compute the
optimal allocation. The objective is to find the optimal allocation for an ex-ante volatility
of around 7%.
Table 12: Expected returns and risks (in %)

µi
σi

(1)
4.2
5.0

(2)
3.8
5.0

(3)
5.3
7.0

(4)
10.4
10.0

(5)
9.2
15.0

(6)
8.6
15.0

(7)
5.3
15.0

(8)
11.0
18.0

(9)
8.8
30.0

Table 13: Correlation matrix of asset returns (in %)

(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)

(1)
100
80
60
−20
−10
−20
−20
−20
0

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

100
40
−20
−20
−10
−20
−20
0

100
50
30
20
20
30
10

100
60
60
50
60
20

100
90
70
70
20

100
60
70
20

100
70
30

100
30

100

Using these figures, we obtain an initial allocation x?(0) that is reported in Table24 14.
The optimal portfolio is invested in only four asset classes. The allocation in US 10Y Bonds
is 28%, while the allocation in High Yield Bonds is 70%. It is obvious that this portfolio
23 This example is taken from Roncalli (2013) on page 287.
24 The weights and the risk/return statistics are given in %.

30

Robust Asset Allocation for Robo-Advisors

cannot be a SAA policy. This is why the Quant will add some constraints in order to obtain
a better solution. We can impose that the weight of one asset class cannot exceed 25%.
Using this new set of constraints Ω(1) , we obtain Portfolio x?(1) that is less concentrated than
Portfolio x?(0) . The allocation in US 10Y Bonds and High Yield Bonds reaches the cap of 25%.
The portfolio is now invested in Euro 10Y Bonds (15.90%), US Equities (10.70%) and EM
Equities (21.27%). The drawback of this solution could be the allocation in equities, which
is too small. This is why the Quant will add another constraint in order to obtain an equity
allocation that is larger than 40%. At the third iteration, we then obtain Portfolio x?(3) . If we
assume that the SAA exercise is complete for a European institutional investor, this solution
is not acceptable because it contains many US assets and too few European assets. This is
why the Quant will add two new constraints. He can require that the allocation in Euro
10Y Bonds is larger than the allocation in US 10Y Bonds, and that the allocation in Euro
Equities is larger than the allocation in US Equities. By using this new set of constraints
Ω(4) , we obtain the following solution: the weight of US 10Y Bonds is 12.13%, the weight
of Euro 10Y Bonds is 22.13%, the weight of IG Bonds is 15.00%, etc. Again, this solution
may not be acceptable, because there is no allocation in Japanese equities. Therefore, the
Quant may impose that there is at least 5% invested in this asset class. After few additional
iterations, the solution is given by the last column in Table 14.
Table 14: The iterative trial-and-error solutions
Step k
US 10Y Bonds
Euro 10Y Bonds
IG Bonds
HY Bonds
US Equities
Euro Equities
Japan Equities
EM Equities
Commodities
µ (x)
σ (x)
SR (x | r)

(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)

#0
28.39
0.00
0.00
69.64
0.00
0.00
0.00
1.17
0.79
8.63
7.00
80.49

#1
25.00
15.90
0.00
25.00
10.70
0.00
0.00
21.27
2.13
7.77
7.00
68.08

#2
24.99
18.60
0.00
16.41
20.86
3.16
0.00
15.98
0.00
7.41
7.00
63.03

#3
25.00
16.50
4.86
10.00
25.00
5.00
0.00
10.00
3.64
7.12
7.00
58.93

#4
12.13
22.13
15.00
10.00
10.00
20.00
0.00
10.00
0.73
6.99
7.00
57.00

···

#K
10.00
30.00
10.00
5.00
10.00
20.00
5.00
8.00
2.00
6.57
6.84
52.17

We notice that the previous iterative process P satisfies:
Ω(k+1) ⊂ Ω(k) ⊂ · · · ⊂ Ω(2) ⊂ Ω(1)
The underlying idea is to define an increasingly constrained investment universe. For instance, we verify that the efficient frontiers are ordered and that they are more and more
constrained (see Figure 9).
Remark 6 Quants may use variants of Problem (21). When they are also in charge of producing µ and Σ, they may also consider the iterative process with the following optimization
problem:
1
x?(k) = arg min x> Σ(k) x − γx> µ(k)
2
In this case, the sequence P is defined as follows:
n
o
P = x?(0) , Ω(1) , Σ(1) , µ(1) , x?(1) , Ω(2) , Σ(2) , µ(2) , x?(2) , . . .
31

Robust Asset Allocation for Robo-Advisors

Figure 9: How does the secret sauce of portfolio optimization work?

10

9

8

7

6

5

4
4

6

8

10

12

It is obvious that the iterative process for defining the optimal portfolio conflicts with an
automated and algorithm-driven robo-advisor. First, this is not the intent of a robo-advisor,
unless we reduce the concept of robo-advisory to a digital application or a data-visualization
tool, meaning that allocation decisions are made outside the robo-advisor. Second, a roboadvisor should be able to manage many portfolios on an industrial scale. If we consider the
traditional lifestyle approach based on three portfolios (defensive, balanced and dynamic),
which are rebalanced at the end of each month, it is obvious that the robo-advisor can
be manually loaded every month. Again, this approach does not correspond to the roboadvisory concept. Indeed, robo-advisors claim that they better meet the expectations of
investors by taking into account their constraints and by being more granular. This is
particularly true with the emergence of goal-based investing in wealth management:
“While mass production has happened a long time ago in investment management through the introduction of mutual funds and more recently exchange
traded funds, a new industrial revolution is currently under way, which involves
mass customization, a production and distribution technique that will allow individual investors to gain access to scalable and cost-efficient forms of goal-based
investing solutions” (Martellini, 2016, page 5).
Lastly, the iterative process does not help improve the portfolio management in a scientific
manner. Indeed, it is a blind-eye approach, because it is difficult to explain the performance
of the portfolio. We don’t know if it comes from the expected returns step (or the active
bets) or the portfolio optimization step. In robo-advisory, these two steps must be easily
identified and distinguished. Indeed, the portfolio optimization engine is part of the roboadvisor while expected returns may be designed outside the robo-advisor. This is generally
the case because they can be imposed by the final investor himself, they can change from
one third-party distributor to another, some investors will want to introduce trend-following
32

Robust Asset Allocation for Robo-Advisors

patterns, etc. Contrary to the optimization method, the engine of expected returns is
therefore not necessarily decided by the fintech that produces the robo-advisor. This is why
the two steps must be perfectly differentiated.

4.2

Formulation of the optimization problem

We note x̃ the reference portfolio25 and xt the current portfolio. The optimized portfolio
for the next period is the solution of this comprehensive optimization program:
x?t+1

=

arg min f (x) + %̃1 Γ̃1 (x − x̃)

2
1
+ %̃2 Γ̃2 (x − x̃) +
2
2
1

(22)

1
2
+%1 kΓ1 (x − xt )k1 + %2 kΓ2 (x − xt )k2
2
 >
 1 x=1
0⩽x⩽1
s.t.

x∈Ω
where Ω is a set of predetermined constraints. This problem considers both L1 and L2
penalty functions with respect to the reference portfolio and the current portfolio. Concerning f (x), we can use the Markowitz function:
f (x) =

1 >
x Σx − γx> µ
2

However, it is certainly better to consider the tracking-error function with respect to the
reference portfolio:
f (x)

=
=

1
>
>
(x − x̃) Σ (x − x̃) − γ (x − x̃) µ
2


1 >
1
>
x Σx − γx
µ + Σx̃ + C
2
γ

where C is a constant that does not depend on the variable x.
The aims of Problem (22) are multiple:
1. The first objective is naturally to optimize the traditional risk/return trade-off.
2. The second objective is to control the active bets between the reference portfolio x̃
and the new optimized portfolio x?t+1 at various levels:
(a) The first layer is to target a tracking error by using the TE objective function in
place of the MVO objective function;
2

(b) The second layer is the L2 penalty %̃2 Γ̃2 (x − x̃) that helps to smooth the
2
tactical allocation with respect to the strategic allocation. This layer implies
shrinking the covariance matrix Σ;
(c) The third layer is the L1 penalty %̃1 Γ̃1 (x − x̃)
that helps to sparsify the
1
relative bets with respect to Portfolio x̃;
3. The third objective is to control the turnover (L1 penalty) and the quadratic costs (L2
penalty) with respect to the current portfolio xt .
25 which is also called the strategic or the benchmark portfolio.

33

Robust Asset Allocation for Robo-Advisors

With all these safeguards, we are equipped to perform stable and robust dynamic allocation for robo-advisors. However, three issues remain unsolved: the specification of expected
returns, the choice of the tracking error level and the calibration of the regularization parameters. The idea of the next section is not to give a solution or to publish our know-how
on these topics (Malongo et al., 2016). However, we will indicate the shortcomings to be
avoided.

4.3

Practical considerations

4.3.1

Incorporating active management views

In some cases, robo-advisors are closed systems, but most of the time, they are open systems. Often, the fintech that developed the robo-advisor technology enters into bilateral
agreements with third-party distributors (asset managers, private banks, wealth managers,
insurance companies, retail distributors, etc.). In this case, the robo-advisor platform is
adapted to take into account the distributor’s specific requirements, constraints and objectives. For instance, the robo-advisor platform may be plugged with the distributor’s
risk/return profiling system. The number of funds and the investment universe changes
from one distributor to another one. One of the big specific features is the engine that
produces expected returns. It is rare that the distributor uses the default engine provided
by the fintech. For instance, some investors will want to incorporate momentum patterns,
others prefer to use expected returns produced by their economic experts, etc.
In practice, it is extremely difficult to express bets in terms of absolute returns. Portfolio
managers prefer to use a rating scale S with different grades. The typical rating scale
contains 7 grades:
Grade
Definition
− − − Strong bearish
−−
Bearish
−
Weak bearish
0
Neutral
+
Weak bullish
++
Bullish
+ + + Strong bullish
The challenge is then to transform these grades into expected returns. The most frequent
empirical approach is based on the Black-Litterman model, which is described in Appendix
B on page 60.
Given a strategic portfolio x̃, we compute the implied expected returns µ̃i of Asset i
thanks to the CAPM equation:
(Σx̃)i
µ̃i = r + SR (x̃ | r) √
(23)
x̃> Σx̃
We assume that the signal si on Asset i is homogeneous to a Sharpe ratio. In particular, we
have:
si
∆ SR i = δ
ns
where ns is the range index of the rating scale26 and δ is a scalar that indicates the flexibility
of active tactical management27 . Then, we deduce that the expected return of the portfolio
26 It is equal to:

ns =

−1 + card S
2

27 Typically, δ is set to one.

34

Robust Asset Allocation for Robo-Advisors

manager is equal to:
µ̆i

=
=

(SR i + ∆ SR i ) · σi
si
µ̃i + δ σi
ns

where SRi = (µ̃i − r) /σi is the implied Sharpe ratio of Asset i relative to the strategic
portfolio x̃ and σi is the estimated volatility of Asset i. The final step is to combine µ̃i and
µ̆i using the Black-Litterman framework:


τ
τ
µi =
µ̃i + 1 −
µ̆i
τ +1
τ +1
where τ is a parameter that measures the confidence into active bets. For instance, when
τ → ∞, the manager’s views are not taken into account, while the conditional expected
returns tends to manager’s views when τ → 0.

Table 15: Covariance matrix of asset classes (Jan. 2016 – Dec. 2016)

(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)

(8)
20.5

(9)
24.3

(10)
17.8

(2)

Volatility (in %)
(4)
(5)
(6)
(7)
7.6
10.1
7.6
16.1
Correlation matrix (in %)
(3)
(4)
(5)
(6)
(7)

(8)

(9)

(10)

100.0
19.4
99.5
2.4
12.7
0.7
2.8
2.0
10.2

100.0
18.1
76.3
87.6
38.1
-4.0
27.6
26.0

100.0
15.6
67.7

100.0
42.9

100.0

(1)
9.2

(2)
7.0

(1)
100.0
17.7
98.1
16.5
71.1
85.9
34.5
-13.2
20.3
16.6

(3)
9.4

100.0
2.1
11.8
1.3
3.6
0.8
10.5

100.0
89.1
68.8
41.0
21.6
57.2

100.0
57.8
18.2
25.3
44.6

100.0
59.5
8.0
54.3

We consider an example with 10 asset classes: (1) US Sovereign Bonds, (2) Euro
Sovereign Bonds, (3) US Investment Grade Bonds, (4) EMU Investment Grade Bonds, (5)
US High Yield Bonds, (6) EM Bonds, (7) US Equities, (8) Europe Equities, (9) Japan Equities and (10) EM Equities. In Table 15, we report the estimated covariance matrix for the
period January 2016 – December 2016. We consider an equally-weighted portfolio x̃, which
corresponds to a 40/60 strategic allocation. By assuming that r = 0 and SR (x̃ | r) = 0.5, we
calculate the vector of implied expected returns using Equation (23). The results are given
in the second column in Table 16. For instance, the implied expected return of US Sovereign
bonds is equal to 2.57%. We now consider a set of manager’s views. The first scenario #1
corresponds to a weak bearish scenario on equity markets. Therefore, the grades are set to
− for the four equity asset classes and + for the two sovereign bond asset classes. In Table
16, we calculate28 the expected returns µ̆ implied by these views, and the final expected
returns µ. For instance, µ̆i and µi are equal to 5.46% and 4.10% for US Sovereign bonds.
We verify that expected returns are increased for sovereign bonds, decreased for equities
and neutral for the other asset classes.
35

Robust Asset Allocation for Robo-Advisors

Table 16: Expected returns in % (scenario #1)
Asset class
US Sov. Bonds
Euro Sov. Bonds
US IG Bonds
EMU IG Bonds
US HY Bonds
EM Bonds
US Equities
Europe Equities
Japan Equities
EM Equities

µ̃i
2.57
0.96
3.02
1.02
4.09
2.88
5.76
6.35
6.76
7.18

si
+
+
0
0
0
0
−
−
−
−

µ̆i
5.64
3.29
3.02
1.02
4.09
2.88
0.40
−0.48
−1.34
1.24

µi
4.10
2.12
3.02
1.02
4.09
2.88
3.08
2.94
2.71
4.21

Table 17: Scenario #2
Asset class
US Sov. Bonds
Euro Sov. Bonds
US IG Bonds
EMU IG Bonds
US HY Bonds
EM Bonds
US Equities
Europe Equities
Japan Equities
EM Equities

µ̃i
2.57
0.96
3.02
1.02
4.09
2.88
5.76
6.35
6.76
7.18

si
0
0
0
0
0
0
+
+++
+
+

µ̆i
2.57
0.96
3.02
1.02
4.09
2.88
11.13
26.85
14.86
13.11

µi
2.57
0.96
3.02
1.02
4.09
2.88
8.45
16.60
10.81
10.14

µ̆i
2.57
0.96
3.02
1.02
4.09
−4.72
5.76
6.35
6.76
−10.62

µi
2.57
0.96
3.02
1.02
4.09
−2.18
5.76
6.35
6.76
−4.69

Table 18: Scenario #3
Asset class
US Sov. Bonds
Euro Sov. Bonds
US IG Bonds
EMU IG Bonds
US HY Bonds
EM Bonds
US Equities
Europe Equities
Japan Equities
EM Equities

µ̃i
2.57
0.96
3.02
1.02
4.09
2.88
5.76
6.35
6.76
7.18

si
0
0
0
0
0
−−−
0
0
0
−−−

36

Robust Asset Allocation for Robo-Advisors

We consider a second scenario that is more favorable to stock markets, in particular
European stocks (see Table 17). By construction, the implied expected returns do not
change because we consider the same strategic allocation. However, the expected returns
µ̆i and µi are different because we have changed the scenario. Finally, we consider a third
scenario in Table 18, which is an adverse scenario on emerging markets29 .
4.3.2

Choosing the right tracking error level

Volatility target strategies are very popular among Quants (Hallerbach, 2012; Hocquard et
al., 2013). This explains why many robo-advisors are based on volatility or tracking error
targeting. As said previously, we prefer TE objective function to MVO objective function.
In this case, there is no constraint on the portfolio volatility, which is related to the volatility
σ (x̃) of the reference portfolio. However, the question of the TE level remains open. We
provide some methods to set the right level of tracking error.
Let x and x̃ be the tactical and strategic portfolios. We have:
σ 2 (x | x̃)

=

σ 2 (Rt (x) − Rt (x̃))

=

σ 2 (x) + σ 2 (x̃) − 2ρ (x, x̃) σ (x) σ (x̃)

where ρ (x, x̃) is the correlation between the portfolio x and the benchmark x̃. Generally,
we have σ (x) ≈ σ (x̃), implying that:
p
(24)
σ (x | x̃) = 2 (1 − ρ (x, x̃)) · σ (x̃)
In Figure 10, we have reported the relationship between the volatility of the strategic portfolio and the tracking error of the portfolio. We notice that it depends on the correlation level.
It follows that if the strategic portfolio’s volatility is low (less than 5%), we cannot target
a high level of tracking error volatility. A level of 1% is certainly the maximum. When the
volatility is moderate between 5% and 10%, we can target a value between 1% and 2%. We
can achieve a higher tracking error only if the portfolio’s volatility is high.
The previous result is of major importance, because it states that the tracking error
level of the tactical portfolio must be related to the volatility of the strategic portfolio. In
practice, the volatility is time-varying, implying that using a constant tracking error strategy
is not optimal.
There is a second reason to consider a time-varying tracking error level, because another
issue concerns the relationship between the tracking error and the active bets. We can show
that (Grinold, 1994):
√
µ (x | x̃) = σ (x | x̃) · TC · IC · n
where TC is the transfer coefficient, IC is the information coefficient and n is the number
of assets. This relationship is known as “the fundamental law of active management”. If
we assume that TC and IC are constant for a given active manager and a given portfolio, it
follows that the excess return is proportional to the tracking error volatility:
µ (x | x̃) ∝ σ (x | x̃)
However, alpha generation is also linked to the number and strength of active bets:
µ (x | x̃) = gµ (s1 , . . . , sn )
28 We assume that δ = 1 and τ = 1.
29 τ is set to 0.5 in order to reflect stronger confidence in this scenario.

37

Robust Asset Allocation for Robo-Advisors

Figure 10: Relationship between volatility and tracking error levels
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0

2

4

6

8

10

12

14

16

18

20

We deduce that the tracking error must be a function of the scores si :
σ (x | x̃) = gσ (s1 , . . . , sn )

(25)

This relationship is essential when considering tactical allocation. Indeed, if all the scores
are equal to zero, there is no active bet, implying that we must target a zero tracking error
level. If all the scores are equal, we are in the same situation. Indeed, since we are bullish
in all the asset classes, there is no reason to deviate from the strategic portfolio. In order
to take a high tracking error risk, we need the bets to present a high dispersion:
si
s1
s2
s3
s4
σ (x | x̃)

#1
0
0
0
0
zero

#2
++
++
++
++
zero

#3
+
−
+
+
moderate

#4
+++
+++
−−−
−−−
high

Since the function gµ is unknown and difficult to estimate, the function gσ is also unknown.
However, we may use the following rule of thumb:


σ (s) + mad (s)
σ (x | x̃) ≈ c ·
· σ+
(26)
2
where σ (s) is the standard deviation of scores, mad (s) is the mean absolute difference of
scores, and σ + is the maximum tracking error. The value of σ + may be deduced from the
relationship (24). By construction, we have:


σ (s) + mad (s)
0≤
≤ 3.6213
2
38

Robust Asset Allocation for Robo-Advisors

and:


0 ≤ lim

n→∞

σ (s) + mad (s)
2


≤3

where n is the number of assets. It follows that the scaling factor c is approximatively equal
1
to .
3
Equation (26) is a preliminary approach to set the level of tracking error. Nevertheless,
this rule of thumb has a major drawback. It does not depend on the asset classes and their
scores. Let us consider the previous example described on page 35. We assume that the
signals are respectively +, 0, +, 0, +, 0, 0, +, 0 and +. In Figure 11, we report the tactical
allocation when we target a tracking error level. For Europe Equities, we have a signal equal
to +, and we verify that the allocation is increasing with respect to the tracking error. For
US Sovereign and IG Bonds, we also have a signal equal to +, but the relationship between
the allocation and the tracking error is not monotonically increasing. The case of US IG
Bonds will be easily solved once we consider Problem (22) instead of a simple tracking
error optimization. The case of US Sovereign Bonds is more problematic. Indeed, in an
initial period when the tracking error is low, the relationship is increasing. However, when
the tracking error increases too much, we obtain the opposite result. The reason is that
the volatility of US Sovereign Bonds is low compared to the other asset classes (equities,
investment grade and high yield). If we increase the tracking error, there is a threshold
beyond which it is better to play only active bets on the most risky assets. Indeed, playing
active bets on low risk assets does not give rise to a high tracking error budget. This is why
the optimizer switches from low-risk assets to high-risk assets. This means that the choice
of a tracking error level depends on the set of parameters: the maximum tracking error that
depends on the strategic portfolio, the scores or active bets and the volatility of the assets
that compose the tactical portfolio.
Figure 11: Relationship between active bets and tracking errors
30

25

20

15

10

5

0
0

1

2

39

3

Robust Asset Allocation for Robo-Advisors

Figure 12: Impact of the ridge parameter on the shrinkage correlation
100
75
50
25

-100

-75

-50

-25

25

50

75

100

-25
-50
-75
-100

4.3.3

Calibrating the regularization parameters

As said previously, the choice of the regularization parameters is not straightforward and
requires a solid expertise and experience. However, we will provide some tips that can help
to calibrate the model30 . The first thing to notice concerns the magnitude of %1 and %2 . On
page 17, we have seen that if Γ̃2 = diag Σ, the regularized correlations are:
ρ̃i,j =

ρi,j
1 + %̃2

In Figure 12, we have reported the relationship between the initial correlation ρi,j and
the shrinkage correlation ρ̃i,j . When %̃2 is equal to zero, ρ̃i,j = ρi,j . When %̃2 → ∞, the
shrinkage correlation tends to zero. We then obtain a diagonal matrix with equal volatilities.
Therefore, there is a trade-off between considering the initial covariance matrix and ignoring
the dependence between assets. A good way to choose %̃2 is to reduce the impact of arbitrage
factors while keeping the significance of common risk factors. If we now consider the L1
penalty %1 kΓ1 (x − xt )k1 and if we set Γ1 = In , the L1 norm measures the portfolio’s twoway turnover:
n
X
k(x − xt )k1 =
|xi − xi,t |
i=1

The parameter %1 may then be used to control the turnover. If Γ1 is a matrix with nonnegative entries that contains the unit transaction costs, the L1 norm measures the portfolio’s
transaction cost (Scherer, 2007). This means that %1 is the average transaction cost if Γ1
is the identity matrix. It follows that the order of magnitude of %̃2 is not comparable to
the order of magnitude of %1 . In the first case, it is expressed as a percentage (for instance,
30 We can also implement cross-validation methods presented in Section 3.6 on page 26

40

Robust Asset Allocation for Robo-Advisors

%̃2 = 25%) whereas in the lasso problem it is expressed in basis points (for instance, %1 = 5
bps). This is in line with the practice that shows that optimal values of L2 regularization are
higher than those of L1 regularization. The second thing to notice concerns the specification
of regularization matrices Γ1 , Γ̃1 , Γ2 and Γ̃2 . Most of the time, they correspond to diagonal
matrices, because it is not easy to consider the cross effects of regularization. The simplest
way is to consider identity matrices, meaning that the regularization patterns reduce to
ridge and lasso approaches. If we use the same parameters %1 = %̃1 and %2 = %̃2 , it is
equivalent to considering that the two portfolios plays a symmetric role. However, this is
not the case. Portfolio xt is used in order to limit the turnover and to smooth the dynamic
allocation. Portfolio x̃ is used in order to control the relative active bets. This is why x̃
is more important than xt for implementing the active management. Last but not least,
the calibration of the parameters highly depends on the investment profile. If the fund
is composed of equities, we need to use more aggressive parameters in order to be more
active than with a multi-asset fund. This means that there is no magic formula, and the
calibration stage requires much empirical research and many tests in order to understand
the interconnectedness between the different terms of the portfolio optimization problem.

5

Conclusion

According to Fisch et al. (2017), robo-advisors are “computer algorithms that provide advice
on investment portfolios and then manage those portfolios”. Since they are digital-based
tools that are generally implemented as web online services, fintechs compete in order to
offer better customization, data visualization, analytics, process automation, etc. And the
concepts of artificial intelligence, big data and machine learning are never far away when
we see the presentation of a robo-advisor. Most of the time, fintechs prefer to insist on the
application’s ergonomics and functionalities, and give little insight into the robo-advisor’s
raison d’être: an automated portfolio allocation engine.
One of the reasons may be that portfolio allocation is more human-based than computerbased. It is true that automation in portfolio optimization is a big issue. Indeed, portfolio
optimization is a hard task and does not always produce the desired results. This is because
the mathematical problem is not necessarily well defined when we would like to obtain a
smooth, sparse, active and dynamic allocation.
In this article, we come back to the traditional mean-variance optimization, and identify
the reason for the issues. We have shown that it primarily corresponds to an alpha optimizer, and not to a beta optimizer. Then we have presented the theory of regularization
and sparsity, and have demonstrated how it improves portfolio optimization. Finally, this
approach is applied for building automated robo-advisory.

41

Robust Asset Allocation for Robo-Advisors

References
[1] Allen, D.M. (1971), Mean Square Error of Prediction as a Criterion for Selecting
Variables, Technometrics, 13(3), pp. 469-475.
[2] Allen, D.M. (1974), The Relationship Between Variable Selection and Data Augmentation and a Method For Prediction, Technometrics, 16(1), pp. 125-127.
[3] Beck, A. (2017), First-Order Methods in Optimization, MOS-SIAM Series on Optimization, 25, SIAM.
[4] Black, F. and Litterman, R.B. (1991), Asset Allocation: Combining Investor Views
with Market Equilibrium, Journal of Fixed Income, 1(2), pp. 7-18.
[5] Black, F. and Litterman, R.B. (1992), Global Portfolio Optimization, Financial
Analysts Journal, 48(5), pp. 28-43.
[6] Boyd, S., Parikh, N., Chu, E., Peleato, B., and Eckstein, J. (2010), Distributed
Optimization and Statistical Learning via the Alternating Direction Method of Multipliers, Foundations and Trends R in Machine learning, 3(1), pp. 1-122.
[7] Broadie, M. (1993), Computing Efficient Frontiers using Estimated Parameters, Annals of Operations Research, 45(1), pp. 21-58.
[8] Brodie, J., Daubechies, I., De Mol, C., Giannone, D., and Loris, I. (2009), Sparse
and Stable Markowitz Portfolios, Proceedings of the National Academy of Sciences,
106(30), pp. 12267-12272.
[9] Bruder, B., Gaussel, N., Richard, J-C., and Roncalli, T. (2013), Regularization
of Portfolio Allocation, SSRN, www.ssrn.com/abstract=2767358.
[10] Candelon, B., Hurlin, C., and Tokpavi, S. (2012), Sampling Error and Double
Shrinkage Estimation of Minimum Variance Portfolios, Journal of Empirical Finance,
19(4), pp. 511-527.
[11] Combettes, P.L., and Müller, C.L. (2018), Perspective Functions: Proximal Calculus and Applications in High-dimensional Statistics, Journal of Mathematical Analysis
and Applications, 457(2), pp. 1283-1306.
[12] Craven, P., and Wahba, G. (1978), Smoothing Noisy Data with Spline Functions,
Numerische Mathematik, 31(4), pp. 377-403.
[13] Diamond, S., Takapoui, R., and Boyd, S. (2018), A General System for Heuristic
Solution of Convex Problems over Nonconvex Sets, Optimization Methods and Software,
33(1), pp. 165-193.
[14] DeMiguel, V., Garlappi, L., Nogales, F.J., and Uppal, R. (2009), A Generalized
Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio
Norms, Management Science, 55(5), pp. 798-812.
[15] DeMiguel, V., Garlappi, L., and Uppal, R. (2009), Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?, Review of Financial Studies,
22(5), pp. 1915-1953.
[16] DeMiguel, V., Martin-Utrera, A., and Nogales, F.J. (2013), Size Matters: Optimal Calibration of Shrinkage Estimators for Portfolio Selection, Journal of Banking &
Finance, 37(8), pp. 3018-3034.
42

Robust Asset Allocation for Robo-Advisors

[17] Duchi, J., Shalev-Shwartz, S., Singer, Y., and Chandra, T. (2008), Efficient
Projections onto the L1 -ball for Learning in High Dimensions, in Cohen, W. (Ed.),
ICML ’08: Proceedings of the 25th International Conference on Machine Learning,
ACM, pp. 272-279.
[18] Engl, H.W., Hanke, M., and Neubauer, A. (1996), Regularization of Inverse Problems, Mathematics and Its Applications, 375, Springer.
[19] Fisch, J.E., Laboure, M., and Turner, (2017), The Economics of Complex Decision
Making: The Emergence of the Robo Adviser, Working paper.
[20] Gabay, D. (1983), Applications of the Method of Multipliers to Variational Inequalities,
in Fortin, M., and Glowinski, R. (Eds), Augmented Lagrangian Methods: Applications
to the Numerical Solution of Boundary-Value Problems, Studies in Mathematics and
Its Applications, 15, pp. 299-331, Elsevier.
[21] Gabay, D., and Mercier, B. (1976), A Dual Algorithm for the Solution of Nonlinear
Variational Problems via Finite Element Approximation, Computers & Mathematics
with Applications, 2(1), pp. 17-40.
[22] Gârleanu, N., and Pedersen, L.H. (2013). Dynamic Trading with Predictable Returns and Transaction Costs, Journal of Finance, 68(6), pp. 2309-2340.
[23] Ghadimi, E., Teixeira, A., Shames, I., and Johansson, M. (2015), Optimal Parameter Selection for the Alternating Direction Method of Multipliers (ADMM): Quadratic
Problems, IEEE Transactions on Automatic Control, 60(3), pp. 644-658.
[24] Giselsson, P., and Boyd, S. (2017), Linear Convergence and Metric Selection for
Douglas-Rachford Splitting and ADMM, IEEE Transactions on Automatic Control,
62(2), pp. 532-544.
[25] Golub, G.H., Heath, M., and Wahba, G. (1979), Generalized Cross-validation as a
Method for Choosing a Good Ridge Parameter, Technometrics, 21(2), pp. 215-223.
[26] Grinold, R.C. (1994), Alpha is Volatility Times IC Times Score, or Real Alphas Don’t
Get Eaten, Journal of Portfolio management, 20(4), pp. 9-16.
[27] Hadamard, J. (1902), Sur les Problèmes aux Dérivées Partielles et Leur Signification
Physique, Princeton University Bulletin, 13, pp. 49-52.
[28] Hallerbach, W.G. (2012), A Proof of the Optimality of Volatility Weighting over
Time, Journal of Investment Strategies, 1(4), pp. 87-99.
[29] Hastie, T., Tibshirani, R. and Friedman, J. (2009), The Elements of Statistical
Learning, Second edition, Springer.
[30] He, B.S., Yang, H., and Wang, S.L. (2000), Alternating Direction Method with SelfAdaptive Penalty Parameters for Monotone Variational Inequalities, Journal of Optimization Theory and applications, 106(2), pp. 337-356.
[31] Hocquard, A., Ng, S., and Papageorgiou, N. (2013), A Constant-volatility Framework for Managing Tail Risk, Journal of Portfolio Management, 39(2), pp. 28-40.
[32] Hoerl, A.E. and Kennard, R.W. (1970), Ridge Regression: Biased Estimation for
Nonorthogonal Problems, Technometrics, 12(1), pp. 55-67.
43

Robust Asset Allocation for Robo-Advisors

[33] Hutchinson, M.F., and De Hoog, F.R. (1985), Smoothing Noisy Data with Spline
functions, Numerische Mathematik, 47(1), pp. 99-106.
[34] Jagannathan, R., and Ma, T. (2003), Risk Reduction in Large Portfolios: Why
Imposing the Wrong Constraints Helps, Journal of Finance, 58(4), pp. 1651-1684.
[35] Jorion, P. (1988), Bayes-Stein Estimation for Portfolio Analysis, Journal of Financial
and Quantitative Analysis, 21(3), pp. 279-292.
[36] Jorion, P. (1992), Portfolio Optimization in Practice, Financial Analysts Journal,
48(1), pp. 68-74.
[37] Laloux, L., Cizeau, P., Bouchaud, J-P., and Potters, M. (1999), Noise Dressing
of Financial Correlation Matrices, Physical Review Letters, 83(7), pp. 1467-1470.
[38] Ledoit, O., and Wolf M. (2003), Improved Estimation of the Covariance Matrix
of Stock Returns With an Application to Portfolio Selection, Journal of Empirical
Finance, 10(5), pp. 603-621.
[39] Ledoit O. and Wolf M. (2004), Honey, I Shrunk the Sample Covariance Matrix,
Journal of Portfolio Management, 30(4), pp. 110-119.
[40] Malongo, H., Lezmi, E., Phung-Que, B., and Roncalli, T., (2016), Asset Allocation for Robo-Advisors, Amundi Internal Research Paper.
[41] Maillard, S., Roncalli, T., and Teı̈letche, J. (2010), The Properties of Equally
Weighted Risk Contribution Portfolios, Journal of Portfolio Management, 36(4), pp.
60-70.
[42] Markowitz, H. (1952), Portfolio Selection, Journal of Finance, 7(1), pp. 77-91.
[43] Markowitz, H. (1956), The Optimization of a Quadratic Function Subject to Linear
Constraints, Naval Research Logistics Quarterly, 3(1-2), pp. 111-133.
[44] Martellini, L. (2016), Mass Customization Versus Mass Production – How an Industrial Revolution is About to Take Place in Money Management and Why It Involves a
Shift from Investment Products to Investment Solutions, Journal of Investment Management, 14(3), pp. 5-13.
[45] Michaud, R.O. (1989), The Markowitz Optimization Enigma: Is ‘Optimized’ Optimal?, Financial Analysts Journal, 45(1), pp. 31-42.
[46] Parikh, N., and Boyd, S. (2014), Proximal Algorithms, Foundations and Trends R in
Optimization, 1(3), pp. 127-239.
[47] OECD (2017), Robo-Advice for Pensions, OECD Report, http://www.oecd.org/
going-digital.
[48] Raghunathan, A.U., and Di Cairano, S. (2014). ADMM for Convex Quadratic
Programs: Q-linear Convergence and Infeasibility Detection, arXiv, 1411.7288.
[49] Richard, J-C., and Roncalli, T. (2015), Smart Beta: Managing Diversification of
Minimum Variance Portfolios, in Jurczenko, E. (Ed.), Risk-based and Factor Investing,,
ISTE Press – Elsevier.
[50] Roncalli, T. (2013), Introduction to Risk Parity and Budgeting, Chapman &
Hall/CRC Financial Mathematics Series.
44

Robust Asset Allocation for Robo-Advisors

[51] Roncalli, T. (2015), Introducing Expected Returns into Risk Parity Portfolios: A
New Framework for Asset Allocation, Bankers, Markets & Investors, 138, pp. 18-28.
[52] Roncalli, T. (2017), Alternative Risk Premia: What Do We Know?, in Jurczenko, E.
(Ed.), Factor Investing and Alternative Risk Premia, ISTE Press – Elsevier.
[53] Scherer, B. (2007), Portfolio Construction & Risk Budgeting, Third edition, Risk
Books.
[54] Sharpe, W.F. (1963), A Simplified Model for Portfolio Analysis, Management Science,
9(2), pp. 277-293.
[55] Sharpe, W.F. (1964), Capital Asset Prices: A Theory of Market Equilibrium under
Conditions of Risk, Journal of Finance, 19(3), pp. 425-442.
[56] Stevens, G.V.G. (1998), On the Inverse of the Covariance Matrix in Portfolio analysis,
Journal of Finance, 53(5), pp. 1821-1827.
[57] Stone, M. (1974), Cross-validation and Multinomial Prediction, Biometrika, 61(3), pp.
509-515.
[58] Stone, M. (1977), An Asymptotic Equivalence of Choice of Model by Cross-validation
and Akaike’s Criterion, Journal of the Royal Statistical Society B, 39(1), pp. 44-47.
[59] Stone, M. (1978), Cross-validation: A Review, Statistics: A Journal of Theoretical
and Applied Statistics, 9(1), pp. 127-139.
[60] Tibshirani, R. (1996), Regression Shrinkage and Selection via the Lasso, Journal of
the Royal Statistical Society B, 58(1), pp. 267-288.
[61] Wahba, G. (1977), Practical Approximate Solutions to Linear Operator Equations
when the Data are Noisy, SIAM Journal on Numerical Analysis, 14(4), pp. 651-667.
[62] Wang, S.L., and Liao, L.Z. (2001), Decomposition Method with a Variable Parameter for a Class of Monotone Variational Inequality Problems, Journal of Optimization
Theory and Applications, 109(2), pp. 415-429.
[63] Zou, H., and Hastie, T. (2005), Regularization and Variable Selection via the Elastic
Net, Journal of the Royal Statistical Society Series B, 67(2), pp. 301-320.

45

Robust Asset Allocation for Robo-Advisors

A

Mathematical results

A.1

Notations

We use the following notations:
• 1Ω (x) is the convex indicator function of Ω: 1Ω (x) = 0 for x ∈ Ω and 1Ω (x) = +∞
for x ∈
/ Ω.
• A† is the Moore-Penrose pseudo-inverse matrix of A; in the scalar case, we have 0† = 0
and a† = a−1 if a 6= 0.
• C = (ρi,j ) denotes the correlation matrix with entries ρi,j .
• Cn (ρ) is the constant correlation matrix of dimension n, whose uniform correlation is
ρ.
• µ is the vector of expected return.
• Σ is the covariance matrix.
Pn
p 1/p
• kxkp = ( i=1 |xi | )
is the Lp norm.
• kxkA = x> Ax

1/2

is the weighted L2 norm.

• [M ]i,j is the (i, j) entry of the matrix M .
• x

y is the Hadamard element-wise product: [x

y]i,j = [x]i,j [y]i,j .

• PΩ (x) is the projection of x on the set Ω:
PΩ (x) = arg miny∈Ω

1
2
ky − xk2
2

• proxf (v) is the proximal operator of f (x):

proxf (v) = arg minx

A.2

1
2
f (x) + kx − vk2
2



Matrix form of the estimators µ̂ and Σ̂

PT
Since we have µ̂ = t=1 wt Rt , it follows that µ̂ = R> w where w = (w1 , . . . , wT ) ∈ RT
and R = (R1 , . . . , RT ) ∈ RT ×n . By noting Dw = diag (w), the expression of the covariance
matrix becomes:
Σ̂

=

T
X

wt Rt Rt> − µ̂µ̂>

t=1

=

T
X
t=1

wt Rt Rt> −

T
X

!
wt Rt

t=1

= R > Dw R − R > w R > w

= R> Dw − ww> R
46

T
X
t=1

>

!>
wt Rt

Robust Asset Allocation for Robo-Advisors

An alternative form is:
Σ̂

T
X

=

>

wt (Rt − µ̂) (Rt − µ̂)

t=1

R − 1µ̂>

>


Dw R − 1µ̂>
>

R − 1w> R Dw R − 1w> R

R> CT> Dw CT R

=
=
=

where CT = IT − 1w> is the (weighted) centering matrix31 . In the case of uniform weights
1
wt = 1/T , CT is equal to IT − 11> . We observe that it is symmetric and idempotent. We
T
1
deduce that Σ̂ = R> CT R.
T

A.3

Relationship between the conditional normal distribution and
the linear regression

Let us consider a Gaussian random vector defined as follows:



 

X
µx
Σxx Σxy
∼N
,
Y
µy
Σyx Σyy
The conditional distribution of Y given X = x is a multivariate normal distribution:

Y | X = x ∼ N µy|x , Σyy|x
where:
µy|x = E [Y | X = x] = µy + Σyx Σ−1
xx (x − µx )
and:
Σyy|x = σ 2 [Y | X = x] = Σyy − Σyx Σ−1
xx Σxy
It follows that Y = µy|x + U where U is a centered Gaussian random variable with variance
s2 = Σyy|x .We recognize the linear regression of Y on X:
Y

= µy + Σyx Σ−1
xx (x − µx ) + U

−1
= µy − Σyx Σ−1
xx µx + Σyx Σxx x + U
= α + β>x + U

−1
where α = µy − Σyx Σ−1
xx µx and β = Σyx Σxx . Moreover, we have:

R2

=
=
=

var (U )
var (Y )
s2
1−
Σyy
1−

Σyx Σ−1
xx Σxy
Σyy

31 We verify that:

CT> Dw CT

because Dw

1 = w and 1> D

=


>


IT − 1w>
Dw IT − 1w>

=

Dw − w1> Dw − Dw 1w> + w1> Dw 1w>

=

Dw − ww>

w 1 = 1.

47

Robust Asset Allocation for Robo-Advisors

Remark 7 In the case where the correlation matrix of the random vector (X, Y ) is constant
– C = Cn+1 (ρ), Maillard et al. (2010) proved that:
−1
Cxx
=

ρ11> − ((n − 1) ρ + 1) In
(n − 1) ρ2 − (n − 2) ρ − 1

We deduce that:
Σ−1 Σxy
xx 
σx
−1
Cxx
Cx,y
=
σy

  
ρ11> − ((n − 1) ρ + 1) In
σx
ρ1
=
σy
(n − 1) ρ2 − (n − 2) ρ − 1

β

=

and:
βi =

ρ (ρ − 1)
σy
·
2
(n − 1) ρ − (n − 2) ρ − 1 σxi

where σy and σx are the standard deviation of random vectors Y and X. The coefficient of
determination becomes:
R2

=

Σyx Σ−1
xx Σxy
Σyy

=

nρ2
nρ − (ρ − 1)

In the two-asset case, we obtain the famous result: R2 = ρ2 . When the number of assets is
very large, the coefficient of determination is equal to the uniform correlation:

1 if ρ < 0
2
lim R =
ρ if ρ ⩾ 0
n→∞

A.4

Tikhonov regularization

We consider the following optimization problem:
x?

=
s.t.

1
1
2
2
kA1 x − b1 k2 + %2 kΓ2 (x − x0 )k2
2
2
A2 x = b2
arg min

(27)

where A1 ∈ RT ×n , b1 ∈ RT ×1 , %2 > 0, Γ ∈ Rn×n , A2 ∈ Rm×n , b2 ∈ Rm×1 and d ∈ Rm×1 .
We assume that A1 has full rank. The Tikhonov matrix Γ2 forces desirable properties of the
solution whereas %2 indicates the strength of the regularization. x0 is an initial solution. In
the case of portfolio optimization, it could be an heuristic portfolio (like the EW portfolio)
or the current allocation in order to control the turnover (Scherer, 2007). The Lagrange
function is equal to:
L (x, λ) =

1
1
2
2
kA1 x − b1 k2 + %2 kΓ2 (x − x0 )k2 + λ> (A2 x − b2 )
2
2

Computation of the gradient leads to:
>
>
∂x L (x, λ) = A>
1 (A1 x − b1 ) + %2 Γ2 Γ2 (x − x0 ) + A2 λ

48

Robust Asset Allocation for Robo-Advisors

Since we have ∂x L (x, λ) = 0 and A2 x = b2 , the optimal portfolio x? is the x-coordinate
solution of the linear system:

 >
  >

>
x
A1 A1 + %2 Γ>
A1 b1 + %2 Γ>
2 Γ 2 A2
2 Γ2 x0
=
(28)
λ
A2
0
b2
This linear system gives the primal and dual variables.

A.5

Limit solutions of Lp - L2 regularization

As %p is fixed and %2 tends to +∞, the formal limit to Problem (18) is given by:
2

x? = arg min kΓ2 (x − x0 )k2

s.t. A2 x = b2

As %2 is fixed and %p tends to +∞, the formal limit to Problem (18) is given by:
p

x? = arg min kΓp (x − x0 )kp

s.t. A2 x = b2

If %p and %2 both tend to +∞, the formal limit to Problem (18) depends on the regime
%p /%2 .

A.6

Augmented QP algorithm

A quadratic programming (QP) problem is an optimization problem with a quadratic objective function and linear constraints:
x?

1
arg min x> A1 x − x> b1
2
s.t. A3 x ⩾ b3
=

(29)

With the inequality constraints, we can easily manage equality constraints and bounds32 . If
we introduce a L2 penalization, the optimization program becomes:
(∗)

=
=

1 >
1
2
x A1 x − x> b1 + %2 kΓ2 (x − x0 )k2
2
2
1 >
1
1
x A1 x − x> b1 + %2 x> Γ2 x − %2 x> Γ2 x0 + %2 x>
0 Γ2 x0
2
2
2

We deduce that the regularization program can be cast into a QP problem:
x?

=
s.t.

1
arg min x> A1 (%2 ) x − x> b1 (%2 )
2
A3 x ⩾ b3

(30)

where A1 (%2 ) = A1 + %2 Γ2 and b1 (%2 ) = b1 + %2 Γ2 x0 .
Let us now introduce an L1 penalization. We have:
x?

where:
f (x) =

=

arg min f (x)

s.t.

A3 x ⩾ b3

1 >
x A1 x − x> b1 + %1 kΓ1 (x − x0 )k1
2

32 An equality constraint A x = b is equivalent to two inequality constraints A x ⩾ b and A x ⩽ b .
2
2
2
2
2
2
The same result applies to bounds x− ⩽ x ⩽ x+ , which can be written as x ⩾ x− and −x ⩾ −x+ .

49

Robust Asset Allocation for Robo-Advisors

and Γ1 is a matrix with non-negative entries. If we use a decomposition of the following
form:
x = x0 + δ + − δ −
(31)

 −
−
+
+
−
−
+
+
with δ = δ1 , . . . , δn , δ = δ1 , . . . , δn , δi ⩾ 0 and δi ⩾ 0, we deduce that:


kΓ1 (x − x0 )k1 = Γ1 δ + − δ − 1 = 1> Γ1 δ + + δ −
The objective function becomes:
f (x) =

1 >
x A1 x − x> b1 + 1> Γ1 δ + + 1> Γ1 δ −
2

Let y = (x, δ − , δ + ) be the vector of unknown variables. We obtain an augmented QP
problem of dimension 3 × n:
y?

1
arg min y > Ã1 y − y > b̃1
2
s.t. Ã3 y ⩾ b̃3
=

where:



A1
Ã1 =  0
0
and:

(32)


0 0
0 0 
0 0




b1

b̃1 =  −Γ>
11
−Γ>
1
1

We can write Equation (31) as follows:
In x + In δ − − In δ + = x0
Since we have δ + ⩾ 0 and δ − ⩾ 0, we deduce that:


A3
0
0
 In
In −In 



In 
Ã3 =  −In −In


0
In
0 
0
0
In
and:




b3
 x0 



b̃3 = 
 −x0 

0 
0

A.7

ADMM algorithm

A.7.1

Dual ascent principle and method of multipliers

The alternating direction method of multipliers (ADMM) is an algorithm introduced by
Gabay and Mercier (1976) to solve problems which can be expressed as33 :
{x? , z ? }

=

arg min f (x) + g (z)

s.t.

Ax + Bz = c

33 We follow the standard presentation of Boyd et al. (2011) on ADMM.

50

(33)

Robust Asset Allocation for Robo-Advisors

where A ∈ Rp×n , B ∈ Rp×m , c ∈ Rp , and the functions f : Rn → R ∪ {+∞} and g : Rm →
R ∪ {+∞} are proper closed convex functions. The expression of the augmented Lagrange
function is:
ϕ
2
Lϕ (x, z, λ) = f (x) + g (z) + λ> (Ax + Bz − c) + kAx + Bz − ck2
2
where ϕ > 0. The ADMM algorithm uses the property that the objective function is
separable, and consists of the following iterations:


x(k+1) = arg min Lϕ x, z (k) , λ(k)



 ϕ
2
(k)
(k)>
(k)
Ax + Bz − c
= arg min f (x) + λ
Ax + Bz − c +
2
2
and:
z (k+1)

=
=



arg min Lϕ x(k+1) , z, λ(k)



 ϕ
2
(k)>
(k+1)
(k+1)
arg min g (z) + λ
Ax
+ Bz − c +
Ax
+ Bz − c
2
2

The update for the dual variable λ is then:


λ(k+1) = λ(k) + ϕ Ax(k+1) + Bz (k+1) − c
We repeat the iterations until convergence.
Boyd et al. (2011) notice that the previous algorithm can be simplified. Let r = Ax +
Bz − c be the (primal) residual. By combining linear and quadratic terms, we have:
λ> r +

ϕ 2
ϕ
ϕ
2
2
r = kr + uk − kuk
2
2
2

where u = ϕ−1 λ is the scaled dual variable. We can then write the Lagrange function (33)
as follows:
Lϕ (x, z, u) = f (x) + g (z) +

ϕ
1
2
2
kAx + Bz − c + uk2 −
kλk
2
2ϕ

Since the last term is a constant, we deduce that the x- and z-updates become:


x(k+1) = arg min Lϕ x, z (k) , u(k)


2
ϕ
= arg min f (x) +
Ax + Bz (k) − c + u(k)
2
2

(34)

(35)

and:
z (k+1)



arg min Lϕ x(k+1) , z, u(k)


2
ϕ
= arg min g (z) +
Ax(k+1) + Bz − c + u(k)
2
2

=

(36)

For the scaled dual variable u(k) , we have:
u(k+1)

= u(k) + r(k+1)


= u(k) + Ax(k+1) + Bz (k+1) − c
51

(37)

Robust Asset Allocation for Robo-Advisors

where r(k+1) = Ax(k+1) + Bz (k+1) − c is the primal residual atiteration k + 1. Boyd et al.
(2011) also defined the variable s(k+1) = ϕA> B z (k+1) − z (k) and refer to s(k+1) as the
dual residual34 at iteration k + 1.
This algorithm benefits from the dual ascent principle and the method of multipliers.
The difference with the latter is that the x and z-updates are performed in an alternating
way. Therefore, it is more flexible because the updates are equivalent to compute proximal
operators for f and g, independently.

A.7.2

Convergence and stopping criteria

Under the assumption that the traditional Lagrange function L0 has a saddlepoint, one
can prove that the residual r(k) converges to zero, the objective function f x(k) + g z (k)
to the optimal value f (x? ) + g (z ? ) , and the dual variable λ(k) to a dual optimal point.
However, the rate of convergence is not known and the primal variables x(k) and z (k) do
not necessarily converge to the optimal values x? and z ? . Nevertheless, in the context of
Markowitz optimization with bound constraints, the results found by Raghunathan and Di
Cairano (2014) may be applied to obtain linear convergence for the primal variables.
In general, the stopping criterion is defined with respect to the residuals:


r(k) 2 ⩽ ε
s(k) 2 ⩽ ε0


where r(k) = Ax(k) + Bz (k) − c and s(k) = ϕA> B z (k) − z (k−1) . Typical values when
implementing this stopping criterion are ε = ε0 = 10−18 .

A.7.3

Penalization parameter and initialization

The convergence result holds regardless of the choice of the penalization parameter ϕ > 0.
But the choice of ϕ affects the speed of convergence (Ghadimi et al., 2015; Giselsson and
Boyd, 2017). In practice, the penalization parameter ϕ may be changed at each iteration,
implying that ϕ is replaced by ϕ(k) and the scaled dual variable uk is equal to λ(k) /ϕ(k) .
This may improve the convergence and make the performance independent of the initial
choice ϕ(0) . To update ϕ(k) in practice, He et al. (2000) and Wang and Liao (2001) provide
a simple and efficient scheme. On the one hand, the x and z-updates in ADMM essentially
2
2
comes from placing a penalty on r(k) 2 . As a consequence, if ϕ(k) is large, r(k) 2 tends to
be small. On the other hand, s(k) depends linearly on ϕ. As a consequence, if ϕ(k) is small,
2
2
2
2
s(k) 2 is small (and r(k) 2 may be large). To keep r(k) 2 and s(k) 2 within a factor µ,
one may consider:

2
2

if r(k) 2 > µ s(k) 2
 τ ϕ(k)
2
2
ϕ(k+1) =
ϕ(k) /τ 0 if s(k) 2 > µ r(k) 2

 (k)
ϕ
otherwise
where µ, τ and τ 0 are parameters that are greater than one. In practice, we use ϕ(0) = 1,
u(0) = 0, µ = 103 and τ = τ 0 = 2.
34 We can interpret s(k+1) as the residual of the dual feasibility conditions: 0 ∈ ∂f (x? ) + A> λ? and
0 ∈ ∂g (z ? ) + B > λ? (Boyd et al., 2011).

52

Robust Asset Allocation for Robo-Advisors

A.7.4

Tikhonov regularization

Let us consider the Tikhonov problem:
x?

1
1
2
2
arg min kA1 x − b1 k2 + %2 kΓ2 (x − x0 )k2
2
2

kxkq ⩽ cq



A2 x = b2
s.t.
A x ⩽ b3


 −3
x ⩽ x ⩽ x+
=

(38)

where q ∈ [1, ∞). We note:
=

Ω2

= {x ∈ Rn : A2 x = b2 }

Ω3

= {x ∈ Rn : A3 x ⩾ b3 }

=
x ∈ R n : x− ⩽ x ⩽ x+

Ω4
We define:
f (x) =

n
o
x ∈ Rn : kxkq ⩽ cq

Ω1

1
1
2
2
kA1 x − b1 k2 + %2 kΓ2 (x − x0 )k2 + 1Ω2 (x)
2
2

and:
g (x) = 1Ω1 (x) + 1Ω3 (x) + 1Ω4 (x)
The Tikhonov problem becomes:
{x? , z ? }

=

arg min f (x) + g (z)

s.t.

x−z =0

Therefore, the ADMM algorithm is:
x(k+1)

=

z (k+1)

=

u(k+1)

=



2
ϕ(k)
arg min f (x) +
x − z (k) + u(k)
2
2


(k)
2
ϕ
(k+1)
(k)
arg min g (z) +
x
−z+u
2
2


u(k) + x(k+1) − z (k+1)

We notice that we can replace the second step by:


z (k+1) = P{g(z)<∞} x(k+1) + u(k)

where P{g(z)<∞} x(k+1) + u(k) is the orthogonal projection of x(k+1) +u(k) onto the convex
set {z ∈ Rn : g (z) < ∞}.With this formulation, the x-step is explicit35 , while the z-step consists in computing orthogonal projections onto a convex set. Explicit formulas for orthogonal
projections are presented in Appendix A.8 on page 55.
35 The x-step is also given by:



>
(k) I
A>
n
1 A1 + %2 Γ2 Γ2 + ϕ
A2

A>
2
0



x(k+1)
λ




=

53

>
(k) z (k) − u(k)
A>
1 b1 + %2 Γ2 Γ2 x0 + ϕ
b2

 

Robust Asset Allocation for Robo-Advisors

A.7.5

Mixed regularization

We now replace the objective function of the Tikhonov problem by:
x? = arg min

1
1
1
2
2
p
kA1 x − b1 k2 + %2 kΓ2 (x − x0 )k2 + %p kΓp (x − x0 )kp
2
2
p

(39)

where p 6= 2. The constraints are the same than those specified for the Tikhonov problem.
We define:
f (x)

=

1
1
2
2
kA1 x − b1 k2 + %2 kΓ2 (x − x0 )k2 +
2
2
1Ω1 (x) + 1Ω2 (x) + 1Ω3 (x) + 1Ω4 (x)

and:
g (z) =

1
p
%p kzkp
p

The L2 − Lp problem becomes:
{x? , z ? }

=

arg min f (x) + g (z)

s.t.

Γp (x − x0 ) − z = 0

With this specification, the ADMM algorithm is:


2
ϕ(k)
x(k+1) = arg min f (x) +
Γp x − z (k) − Γp x0 + u(k)
2
2


(k)
2
ϕ
z (k+1) = arg min g (z) +
Γp x(k+1) − z − Γp x0 + u(k)
2
2


u(k+1) = u(k) + Γp x(k+1) − z (k+1) − Γp x0
The x-step consists in minimizing a quadratic constrained problem. It can be carried out
explicitly if no inequality constraint is imposed. Otherwise, the x-step can be performed by
p
another ADMM. The z-step consists in computing the proximal operator of λ kzkp at the

point z = Γp x(k+1) − Γp x0 + u(k) with λ = %p / pϕ(k) . Other choices for the functions f (x)
and g (z) give rise to computing constrained proximal operators or the proximal operator of
p
x 7→ kΓp xkp . No explicit formula is known for the latter, unless a positive multiple of Γp
is orthogonal (Beck, 2017). Our choice makes the z-step explicit for p ∈ {1, 2, 3, 4, 5}, and
easily computable for any p > 1.
A.7.6

Cardinality constraints

The ADMM algorithm can also be used to find a portfolio with at most n1 non-zero weights.
Let us introduce the set Z of n1 -sparse vectors:

Z = x ∈ Rn | card x ⩽ n1 , x− ⩽ x ⩽ x+
(40)
We consider the augmented Tikohnov problem:
x?

1
1
2
2
arg min kA1 x − b1 k2 + %2 kΓ2 (x − x0 )k2
2
2

x ∈ Ω1 ∩ Ω2 ∩ Ω 3 ∩ Ω 4
s.t.
Γ1 (x − x0 ) ∈ Z
=

54

(41)

Robust Asset Allocation for Robo-Advisors

Zou and Hastie (2005) have been introduced Problem (39) with p = 1 as a convex relaxation
to problem (41). The constraint x ∈ Z is forced by the penalty %1 kΓ1 (x − x0 )k1 and the
strength of the penalty parameter %1 must be chosen as the smallest value that satisfies the
constraint card x ⩽ n1 (Hastie et al., 2009).
The projection onto the non-convex set Z exists and is explicit (but may not be unique).
Diamond et al. (2018) show that:
PZ (v) = PΩ4 (v (n1 ))
where v (n1 )i = vi if i ∈ I, v (n1 )i = 0 if i 6∈ I, I is a set of indices of the n1 largest values of
|vi |, and PΩ4 is the projection onto Ω4 = {x ∈ Rn : x− ⩽ x ⩽ x+ }. As previously, we have:
f (x)

=

1
1
2
2
kA1 x − b1 k2 + %2 kΓ2 (x − x0 )k2 +
2
2
1Ω1 (x) + 1Ω2 (x) + 1Ω3 (x) + 1Ω4 (x)

and:
g (z) = 1Z (z)
with the constraint Γ1 (x − x0 ) = z. With this specification, the ADMM algorithm is:


2
ϕ(k)
(k+1)
(k)
(k)
x
= arg min f (x) +
Γp x − z − Γp x0 + u
2
2


z (k+1) = PZ Γ1 x(k+1) − Γ1 x0 + u(k)


u(k+1) = u(k) + Γ1 x(k+1) − z (k+1) − Γ1 x0
Hence, the z-step is explicit. The ADMM does not necessarily converge, and when it does, it
does not necessarily converge to an optimal point. Contrary to the convex case, the possible
convergence of the algorithm depends on the initial values of x0 and the penalization parameter ϕ(k) . In the non-convex setting, the ADMM may be considered as a local optimization
method, and local neighbor search method with convex relaxation and restrictions may be
used to obtain the convergence of the algorithm (Diamond et al., 2018).

A.8

Proximal operators and projections

As shown previously, the z-step of the ADMM algorithm generally computes the proximal
operator of a norm or the projection onto the intersection of simple convex sets. We review
the most useful cases in active asset management and we refer the reader to Parikh and
Boyd (2014), Beck (2017), and Combettes and Müller (2018) for further examples. In most
of these cases, the proximal operators are explicit or consists in determining the zero of a
real-valued function.
A.8.1

Definition of the proximal operator

Let f : Rn → R ∪ {+∞} be a proper closed convex function. The proximal operator
proxf (v) : Rn → Rn is defined by:


1
2
(42)
proxf (v) = x? = arg minx f (x) + kx − vk2
2
1
2
kx − vk2 is strongly convex, it has a unique minimum
2
for every v ∈ Rn (Beck, 2017; Parikh and Boyd, 2014).
Since the function fv (x) = f (x) +

55

Robust Asset Allocation for Robo-Advisors

If we would like to compute the proximal operator of λf (x) + 1Ω (x) for λ ⩾ 0, one 

[The evaluation harness truncated this reference: showing the first 120000 of 133314 characters.]
</reference>

<statements>
1. The evidence supports a qualified yes: hybrids work best when they assign each model family the job it is good at, but the strongest hybrid forms remain research-stage, while the available production evidence still describes regularized mean-variance and Black-Litterman workflows rather than learned allocators.
2. Roncalli adds a structural explanation: mean-variance optimization focuses on small eigenvalues of the covariance matrix, which is why stability is lacking, and small changes in volatilities or correlations can materially change weights.
3. Roncalli states that raw mean-variance solutions are generally not satisfied in practice, so practitioners spend substantial time adding and testing constraints, and that constraint-based regularization introduces the personal views of the quant in charge and requires repeated trial and error
4. Roncalli notes that Black-Litterman is slightly more robust than Markowitz because the benchmark or tracking-error structure is already a form of regularization, but because Black-Litterman is a slight modification of Markowitz, it suffers from the same drawbacks
5. He also warns that certain Black-Litterman parameterizations can dramatically reduce the covariance matrix; as tau approaches zero, the conditional expectation tends toward the manager’s views while the covariance tends toward the null matrix, creating an arbitrage between reliance on subjective views and loss of variance information
6. Roncalli records that the Markowitz approach became the standard model for strategic asset allocation until the end of the 2000s, and that since the 2008 crisis risk budgeting, including risk parity and equal risk contribution, has emerged as a serious contender because it produces stable portfolios without relying on expected returns
7. For tactical asset allocation, professionals generally prefer Black-Litterman because the optimized portfolio depends on the current allocation, but the model remains vulnerable to the same mean-variance drawbacks
8. The documented robustification routes are input estimation methods such as de-noising and covariance shrinkage, and objective-function regularization such as L1 or L2 penalties, which in robo-advisor settings require non-traditional algorithms based on ADMM and proximal operators
9. Roncalli also notes that the rise of robo-advisors changes the trend by creating demand for active allocation models focused on expected returns, because the challenge is tactical dynamic allocation rather than static strategic profiling
10. Roncalli’s robo-advisor study, published in 2019, describes deployed practice as regularized mean-variance and Black-Litterman with denoising, shrinkage, weight constraints, and norm penalties, and notes that risk budgeting became a serious contender after 2008 because it avoids expected-return estimation
11. The paper’s projection that robo-advisors would manage around $1 trillion by 2020 is a historical estimate, not a 2026 observation
12. Mean-variance optimization measures risk as variance, requires estimates of expected returns and covariances, and produces a transparent but often unstable weight vector; its original formulation does not assume normality, but it ignores higher moments and is highly sensitive to estimation error.
13. Black-Litterman measures risk through the same mean-variance lens unless extended, obtains expected returns from a market-equilibrium prior plus subjective views, and produces market-anchored tilts that are more intuitive but still view-dependent and still vulnerable to mean-variance fragility.
14. The most defensible framework combines a stable equilibrium or regularized mean-variance backbone
15. What would change the judgment is not another backtest with a higher Sharpe ratio, but long-sample out-of-sample results after realistic costs, robustness across market regimes and holding horizons, transparent attribution that survives auditability, and deployment evidence showing that learned or hybrid allocators displace regularized mean-variance and Black-Litterman workflows in live robo-advisor or institutional asset-allocation settings.
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.